## Fun teaching resources & tips to help you teach math with confidence

## Problem Solving by Working Backwards

As I’ve shared before, there are many different ways to go about solving a math problem, and equipping kids to be successful problem solvers is just as important as teaching computation and algorithms. In my experience, students’ frustration often comes from not knowing where to start. Providing them with strategies enables them to at least get the ideas flowing and hopefully get some things down on paper. As in all areas of life, the hardest part is getting started! Today I want to explain how to teach problem solving by working backwards .

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–>Pssst! Do your kids need help making sense of and solving word problems? You might like this set of editable word problem solving templates ! Use these with any grade level, for any type of word problem :

## Solve a Math Problem by Working Backwards:

Before students can learn to recognize when this is a helpful strategy, they must understand what it means. Working backwards is to start with the final solution and work back one step at a time to get to the beginning.

To help show your students what this looks like, you might start by thinking about directions. Write out some basic directions from home to school:

- Start: Home
- Turn right on Gray St.
- Turn left on Sycamore Ln.
- Turn left on Rose Dr.
- Turn right on Schoolhouse Rd.
- End: School

Ask students to then use this information to give directions from the school back home . Depending on the age of your students, you may even want to draw a map so they can see clearly that they have to do the opposite as they make their way back home from school. In other words, they need to “undo” each turn to get back, i.e. turn left on Schoolhouse Rd. and then right on Rose Dr. etc.

In math, these are called inverse operations . When using the “work backwards” strategy, each operation must be reversed to get back to the beginning. So if working forwards requires addition, when students work backwards they will need to subtract. And if they multiply working forwards, they must divide when working backwards.

Once students understand inverse operations, and know that they must start with the solution and work back to the beginning, they will need to learn to recognize the types of problems that require working backwards.

In general, problems that list a series of events or a sequence of steps can be solved by working backwards.

## Here’s an example:

Sam’s mom left a plate of cookies on the counter. Sam ate 2 of them, his dad ate 3 of them and they gave 12 to the neighbor. At the end of the day, only 4 cookies were left on the plate. How many cookies did she make altogether?

In this case, we know that the final cookie amount is 4. So if we work backwards to “put back” all the cookies that were taken or eaten, we can figure out what number they started with.

Because cookies are being taken away, that denotes subtraction. Thus, to get back to the original number we have to do the opposite: add. If you take the 4 that are left and add the 12 given to the neighbors, and add the 3 that Dad ate, and then add the 2 that Sam ate, we find that Sam’s mom made 21 cookies .

You may want to give students a few similar problems to let them see when working backwards is useful, and what problems look like that require working backwards to solve.

Have you taught or discussed problem solving by working backwards with your students? What are some other examples of when this might be useful or necessary?

## Don’t miss the other useful articles in this Problem Solving Series:

- Problem Solve by Drawing a Picture
- Problem Solve by Solving an Easier Problem
- Problem Solve with Guess & Check
- Problem Solve by Finding a Pattern
- Problem Solve by Making a List

## Never Run Out of Fun Math Ideas

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So glad to have come across this post! Today, word problems were the cause of a homework meltdown. At least tomorrow I’ll have a different strategy to try! #ThoughtfulSpot

I’m so glad to hear that! I hope you found some useful ideas!! Homework meltdowns are never fun!! Best of luck!

This is really a great help! We have just started using this method for some of my sons math problems and it helps loads. Thanks so much for sharing on the Let Kids Be Kids Linkup!

That’s great Erin! I hope this is a helpful method and makes things easier for your son! 🙂

I’ve not used this method before but sounds like a good resource to teach. Thanks for linking #LetKidsBeKids

I hope this proves to be helpful for you!

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## Why the Most Forward-Thinking Product Teams Work Backwards

You’ve probably heard the old adage “success is not a destination, but a journey.” Clearly its authors were not product managers. Product teams aren’t measured on the winding road and applauded for the many steps taken—it’s all about results. Yet, we’re often so consumed with the process, routine, and endless cycle of building that we can forget none of that matters if the end result doesn’t meet our goals.

That’s why the best plans start with the final outcome. Productivity gurus FranklinCovey even trademarked this as one of their seven habits: Begin with the end in mind.

Many innovative companies have adopted this working backwards mindset to keep people from being distracted during the development process and maintain a singular focus on the ultimate goal.

For example, Amazon starts every project with the product manager authoring a press release articulating the current problem. They explore why other solutions aren’t cutting it, and how they’re solving things for the customer with their not-yet-built product. Throughout the entire product development process, anyone can refer back to that document as a guidepost, ensuring everything they do is making that press release a reality.

## What is working backwards?

Working backwards is all about starting with the desired end result in mind and then figuring out how to get there. Although everyone should already do this, there are plenty of times when this isn’t the case.

The key to working backwards is crystalizing where you want to end up. This means diving into the details of what the desired end result should be (more sales, happier customers, easier workflows, etc.), what the product must be capable of to achieve those goals, and what success would look like.

Looking at Apple’s slick and popular product line, it’s obvious they were listening when Steve Jobs said “you have to start with the customer experience and work backwards towards the technology.”

With an agreed upon vision, the product team can then break down all of the required steps to reach that final destination. This differs from taking a more exploratory, iterative, “let’s see where this takes us” approach, or simply building stuff for the sake of building it.

## Why the best product teams work backwards

Working backwards creates a more efficient and focused product development process. With a clearly articulated endpoint in mind, teams don’t get distracted or build the wrong things as they fumble forward. Instead, they know exactly what the finished product must be capable of, so they’re far more likely to get it right the first time.

This approach also prevents the team from getting lost in the weeds and worrying about implementation instead of results. As we’ve heard many times from our engineering counterparts: “Don’t tell me HOW to build it, just tell me WHAT to build.”

When you’re working backwards, the WHAT has already been very well-defined and no one will have any questions about whether the final product meets expectations. This not only gives clear guidelines to product development; it also provides QA with precise things to look for during their testing and gives a head start for sales and marketing since they know what they’re getting.

Prioritization also benefits from this approach; if a particular item doesn’t move the product closer to the desired finish line, then it takes a backseat to the items that do. It hones everyone in on satisfying customers and delivering value accordingly.

## How to get your team to work backwards

Working backwards may be a state of mind, but getting the rest of the team on board might require something a little more tangible to wrap their heads around. Here are three methods for hitting “rewind” instead of “fast forward.”

## Starting with a press release

Made famous by Amazon, the working backwards strategy is a favorite among many product teams and thought leaders. A press release is usually the very last step in the product development and launch process. It tells the world: “Here I am, this is what I can do, and this is why you should care.”

To be effective, the author must step back from the technobabble trap and communicate in terms that resonate with the target customer.

“One important element of the press release is that it is written in so-called ‘Oprah-speak’. Or in other words, a way that is easy to understand,” says Nikki Gilliard of Econsultancy. “This essentially allows Amazon to cut through tech-jargon and any descriptions that would only confuse the customer, in order to deliver a mainstream product.”

The starting point for the product definition is a customer-centric document, unconcerned with implementation details, technology or user interface design. Then, the focus shifts to what encompasses a truly great solution for the customer. If the press release is compelling, then you’re onto something.

“Iterating on a press release is a lot quicker and less expensive than iterating on the product itself,” says Amazon’s Ian McAllister . “If the press release is hard to write, then the product is probably going to suck. Keep working at it until the outline for each paragraph flows.”

But struggling to get the press release right is part of the process—if you could bang it out in an afternoon then you haven’t done the homework and made it bulletproof enough to drive an entire product development cycle.

“I created a couple of them in the past and it took me a lot of time; several weeks or even months actually, as the amount of time you need to dedicate on research is high, and trying to explain your idea like you would to real customers requires a lot of effort and dedication,” says Andrea Marchiotto of Unilever.

And because the press release is intended to declare the company’s success with the product—not just its availability—there must be a significant business case for it as well, which matches Amazon’s approach to selecting new features. For example, 90% of Amazon Web Services roadmap is driven by broad customer requests , indicating advance knowledge of a clear demand.

## Conducting a pre-mortem

A post-mortem (or after-the-fact review of everything that went right or wrong during a project) is a common method for organizations to learn from previous mistakes and successes to perform even better the next time around. It’s a group affair where representatives from the entire organization chime in on what worked well and what went astray.

A pre-mortem essentially places that same group of stakeholders in a time machine and asks them to imagine everything that could happen before a single line of code is written or design is mocked up. The goal is anticipating and wargaming the situation, spotting all possible scenarios so the team has already anticipated potential stumbling blocks and land mines.

These sessions begin by brainstorming every possible calamity that could befall your product, from total market rejection to compromising user data to sluggish performance and inability to scale. It’s a chance to surface every fear and doubt lurking in people’s minds and determining which are more likely to actually occur.

“Once you’ve collectively established your highest risks, you can start thinking about ways to mitigate these risks. Realistically, you might not be able to stop all risks from happening,” says Marc Abraham of Settled. “In these scenarios, you can still figure out how to best reduce the impact of a risk happening and come up with a ‘plan B’.”

If the mitigation strategy isn’t obvious, a sub-team can be assigned to each outstanding item. Then, the team can work out how to deal with it if it arises (or fully preventing it from occurring at all). And with all potential horror show endings in view, the product team can work backwards minimizing or avoiding as much as possible.

Believe it or not, this might even serve as a bonding exercise for the team, forced to identify and grapple with possible unpleasantness and tap their problem-solving skills.

## Begin with your Product Hunt page

Similar to the press release tactic, this method also requires the product team to identify its ideal output and work backwards from there.

Product Hunt is one of the leading showcases for new products and a semi-meritorious platform for building buzz and traffic from early adopters. A Product Hunt page includes a 60-character-max tagline, a thumbnail image, a gallery, a two-sentence description, three or four topics that your product fits into and then finally a “maker comment,” which according to the Product Hunt blog should:

“Briefly introduce yourself, the team, and the problem that you’re solving. In a total of 3–4 sentences explain what the value prop is, what’s the use case, who its for, and why you are building it. If this is the second time you’re launching on PH (i.e. a big product update or huge feature announcement), explain what’s changed. Tip: Make it as easy as possible for people to care.”

In total, you’ve got about seven or eight sentences and a few visuals to communicate what your product does, who would want to use it, why they should use it and what makes your product and team so special. Distilling your grand product vision down to this tiny bit of text is not only a great exercise in editing and brevity, but it also forces the team to really lock in on what’s most important.

With your faux Product Hunt page already authored, this artifact can be used to gain consensus in the organization and serve as ongoing inspiration during the product development. It’s conciseness and focus on the customer experience and value proposition is a great point to work backwards from.

Regardless of how you bring a working backwards mentality to your product team, the ingredient pulling everything together is truly understanding what customers need before you begin working on everything. If you haven’t done your homework in that department then your perfect final product may miss the mark, leading to a post-mortem that doesn’t do the product team any favors.

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## 2.5.3: Guess and Check, Work Backward

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## Guess and Check, Work Backward

Suppose that you and your brother both play baseball. Last season, you had 12 more hits than 3 times the number of hits that your brother had. If you had 159 hits, could you figure out how many hits your brother had?

## More Problem Solving Strategies

This lesson will expand your toolbox of problem-solving strategies to include guess and check and working backward . Let’s begin by reviewing the four-step problem-solving plan:

Step 1: Understand the problem.

Step 2: Devise a plan – Translate.

Step 3: Carry out the plan – Solve.

Step 4: Look – Check and Interpret.

## Develop and Use the Strategy: Guess and Check

The strategy for the “guess and check” method is to guess a solution and use that guess in the problem to see if you get the correct answer. If the answer is too big or too small, then make another guess that will get you closer to the goal. You continue guessing until you arrive at the correct solution. The process might sound like a long one; however, the guessing process will often lead you to patterns that you can use to make better guesses along the way.

## Let's use the guess and check method to solve the following problem:

Nadia takes a ribbon that is 48 inches long and cuts it in two pieces. One piece is three times as long as the other. How long is each piece?

We need to find two numbers that add to 48. One number is three times the other number.

Guess 5 and 15. The sum is 5+15=20, which is too small.

Guess bigger numbers 6 and 18. The sum is 6+18=24, which is too small.

However, you can see that the previous answer is exactly half of 48.

Multiply 6 and 18 by two.

Our next guess is 12 and 36. The sum is 12+36=48. This is correct.

## Develop and Use the Strategy: Work Backward

The “work backward” method works well for problems in which a series of operations is applied to an unknown quantity and you are given the resulting value. The strategy in these problems is to start with the result and apply the operations in reverse order until you find the unknown. Let’s see how this method works by solving the following problem.

## Let's solve the following problem by working backwards :

Anne has a certain amount of money in her bank account on Friday morning. During the day she writes a check for $24.50, makes an ATM withdrawal of $80, and deposits a check for $235. At the end of the day, she sees that her balance is $451.25. How much money did she have in the bank at the beginning of the day?

We need to find the money in Anne’s bank account at the beginning of the day on Friday. From the unknown amount, we subtract $24.50 and $80 and we add $235. We end up with $451.25. We need to start with the result and apply the operations in reverse.

Start with $451.25. Subtract $235, add $80, and then add $24.50.

451.25−235+80+24.50=320.75

Anne had $320.75 in her account at the beginning of the day on Friday.

## Plan and Compare Alternative Approaches to Solving Problems

Most word problems can be solved in more than one way. Often one method is more straightforward than others. In this section, you will see how different problem-solving approaches compare when solving different kinds of problems.

## Now, let's solve the following problem by using the both the guess and check method and the working backward method:

Nadia’s father is 36. He is 16 years older than four times Nadia’s age. How old is Nadia?

This problem can be solved with either of the strategies you learned in this section. Let’s solve the problem using both strategies.

Guess and Check Method:

We need to find Nadia’s age.

We know that her father is 16 years older than four times her age, or 4× (Nadia’s age) + 16.

We know her father is 36 years old.

Work Backward Method:

Nadia’s father is 36 years old.

To get from Nadia’s age to her father’s age, we multiply Nadia’s age by four and add 16.

Working backward means we start with the father’s age, subtract 16, and divide by 4.

Example 2.5.3.1

Earlier, you were told that you had 12 more hits than 3 times the number of hits that your brother had. If you had 159 hits, how many hits did your brother have?

Since we know how many hits you had, we can work backward to determine the number of hits that your brother had.

Because you had 12 more hits than 3 times the number of hits that your brother had, we do the opposite: subtract 12 and divide by 3.

159−12=147

147÷3=49

Your brother had 49 hits.

Example 2.5.3.2

Hana rents a car for a day. Her car rental company charges $50 per day and $0.40 per mile. Peter rents a car from a different company that charges $70 per day and $0.30 per mile. How many miles do they have to drive before Hana and Peter pay the same price for the rental for the same number of miles?

Hana’s total cost is $50 plus $0.40 times the number of miles.

Peter’s total cost is $70 plus $0.30 times the number of miles.

Guess the number of miles and use this guess to calculate Hana’s and Peter’s total cost.

Keep guessing until their total cost is the same.

Guess 50 miles.

Check $50+$0.40(50)=$70 $70+$0.30(50)=$85

Guess 60 miles.

Check $50+$0.40(60)=$74 $70+$0.30(60)=$88

Notice that for an increase of 10 miles, the difference between total costs fell from $15 to $14. To get the difference to zero, we should try increasing the mileage by 140 miles.

Guess 200 miles

Check $50+$0.40(200)=$130 $70+$0.30(200)=$130correct

- Nadia is at home and Peter is at school, which is 6 miles away from home. They start traveling toward each other at the same time. Nadia is walking at 3.5 miles per hour and Peter is skateboarding at 6 miles per hour. When will they meet and how far from home is their meeting place?
- Peter bought several notebooks at Staples for $2.25 each and he bought a few more notebooks at Rite-Aid for $2 each. He spent the same amount of money in both places and he bought 17 notebooks in total. How many notebooks did Peter buy in each store?
- Andrew took a handful of change out of his pocket and noticed that he was holding only dimes and quarters in his hand. He counted that he had 22 coins that amounted to $4. How many quarters and how many dimes does Andrew have?
- Anne wants to put a fence around her rose bed that is one-and-a-half times as long as it is wide. She uses 50 feet of fencing. What are the dimensions of the garden?
- Peter is outside looking at the pigs and chickens in the yard. Nadia is indoors and cannot see the animals. Peter gives her a puzzle. He tells her that he counts 13 heads and 36 feet and asks her how many pigs and how many chickens are in the yard. Help Nadia find the answer.
- Andrew invests $8000 in two types of accounts: a savings account that pays 5.25% interest per year and a more risky account that pays 9% interest per year. At the end of the year, he has $450 in interest from the two accounts. Find the amount of money invested in each account.
- There is a bowl of candy sitting on our kitchen table. This morning Nadia takes one-sixth of the candy. Later that morning Peter takes one-fourth of the candy that’s left. This afternoon, Andrew takes one-fifth of what’s left in the bowl and finally Anne takes one-third of what is left in the bowl. If there are 16 candies left in the bowl at the end of the day, how much candy was there at the beginning of the day?
- Nadia can completely mow the lawn by herself in 30 minutes. Peter can completely mow the lawn by himself in 45 minutes. How long does it take both of them to mow the lawn together?

Mixed Review

- Rewrite √500 as a simplified square root.
- To which number categories does −2/13 belong?
- Simplify 1/2|19−65|−14.
- Which property is being applied? 16+4c+11=(16+11)+4c
- Is {(4,2),(4,−2),(9,3),(9,−3)} a function?
- Write using function notation: y=(1/12)x−5.
- Jordyn spent $36 on four cases of soda. How much was each case?

## Review (Answers)

To see the Review answers, open this PDF file and look for section 2.14.

## Additional Resources

Activity: Guess and Check, Work Backward Discussion Questions

Practice: Guess and Check, Work Backward

Real World Application: Car Loan

## Working Backwards: Strategic Approach to Challenges

In the fast-paced and ever-evolving world we inhabit, challenges are as constant as the ticking of a clock. The Working Backwards Strategy is a beacon of hope amidst the formidable waves of complexity that confront individuals and organizations alike. In essence, this strategic approach is a clarion call to begin where most end – the desired outcome. By flipping conventional problem-solving on its head, this strategy empowers us to navigate through the murky waters of uncertainty and forge a clear path towards our goals.

The relevance of the Working Backwards Strategy can hardly be overstated - it imparts a robust framework to individuals and enterprises for overcoming challenges that may at first appear insurmountable. This article illuminates the intricate cogs and wheels that constitute the strategic approach to problem-solving, thereby equipping readers with an arsenal of tactics to conquer the obstacles that beset their paths.

As we embark on this enlightening journey through the annals of the Working Backwards Strategy , we will dissect its historical underpinnings, explore its mechanics, and glean insight from real-world applications. This expedition promises not just theoretical knowledge but also practical wisdom, aiding our understanding of how this approach dovetails seamlessly with innovative thinking to produce remarkable results.

## The Genesis of Working Backwards Strategy

The Working Backwards Strategy is more than a mere tactic; it is a paradigm shift that has its roots firmly planted in a historical context. Its origin can be traced back to the stoic philosophers, who often emphasized the significance of visualizing the end to make sense of the present. Fast forward to the modern era, and we observe that this strategy has been honed and tailored to address the complexities of contemporary problems, particularly in business and technology.

Innovation has become the lifeblood of today's corporate giants and disruptive startups. The Working Backwards Strategy is paramount in this arena as it encourages an outcome-oriented outlook. When the endpoint is clear and immovable, the path to reach it can be crafted with precision and agility, aligning perfectly with the demands for creative and divergent thinking.

Not only does the strategy align with the forces of change propelling today's markets, but it also synergizes with the kind of innovative thinking that paves the way for groundbreaking advances. Even in a world that venerates forward-thinking, looking backwards from the future to the present offers the clarity needed to cut through the noise and home in on what truly matters.

## Understanding the Mechanics of Working Backwards

Central to the Working Backwards Strategy are several core principles that guide its disciplined application. This strategy necessitates a departure from linear planning and proffers a more cyclic and iterative model of strategy development. It asks us to start with the customer experience and work our way to the minimum set of creative solutions needed to meet that experience.

## Step 1: Identifying the ideal end-state

The inception of this journey demands a clear definition of the ultimate goal. This vision serves as a lighthouse, guiding all subsequent decisions and strategies. By envisioning the ideal end-state, one can crystallize their objectives and garner insight into the expectations that need fulfillment. Understanding the final destination is crucial as it underpins the entire process that follows.

## Step 2: Establishing key milestones

Once a clear endpoint is discerned, the next phase involves charting out significant milestones. These act as checkpoints that help track progress and ensure that all efforts are congruent with the ultimate goal. Milestones define the critical junctures in the journey to success and lay out the high-level strategy that must be followed.

## Step 3: Creating a reverse action plan

The creation of a reverse action plan is where the strategy takes tangible form. This plan draws directly from the identified milestones and sketches a backward path to the present. Each step is scrutinized through the lens of necessity and efficaciousness, with the primary criterion being its contribution to reaching the end goal.

This tactical phase is also where innovative thinking comes into full play. Understanding that the desired future state might require unprecedented solutions, practitioners of this strategy are encouraged to adopt an explorative mindset, entertain unconventional ideas, and remain adaptable in their approach.

## Application in Various Contexts

Overcoming challenges in business development.

Example: Product development in tech companies

In the realm of technology, the strategy has proven instrumental in navigating the turbulent waters of product development. Take, for instance, the creation of a new software platform. Working Backwards entails envisioning the complete, seamless user experience before writing a single line of code. By focusing on the desired utility and ease of use, tech companies can reverse engineer the process to ensure each development phase contributes directly to the end user's needs.

## Example: Marketing strategies in startups

For startups, crafting a marketing strategy can be a daunting task with resource limitations. Working Backwards from the client acquisition targets can help startups prioritize high-impact marketing efforts and optimize the allocation of their limited resources. This approach allows them to carve a niche in the market while ensuring sustainability and growth.

## Problem-Solving Techniques in Personal Goals

Example: Career advancement planning

Working Backwards is not confined to the corporate and technological spheres - it's equally applicable to personal development areas such as career planning. By starting with the end goal of a desired job position, one can delineate the certifications, skills, and experiences required to achieve it. Opting for a problem solving course or an online mba course might emerge as pivotal steps in this tailored plan.

## Example: Educational achievements

Students aiming for high academic achievement can also benefit from this strategy. By picturing their names on the dean's list, they can craft a detailed study schedule that revolves around this goal, carefully selecting the courses and extracurricular activities that align with their academic aspirations.

The Working Backwards Strategy has cemented itself as an essential component of the problem-solver's toolkit, offering a robust and strategic scaffold for surmounting challenges across a spectrum of domains. In reflecting on its utility, the strategy's penchant for fostering innovative thinking and meticulous goal orientation stands out as particularly compelling.

As we draw this exploration to a close, it becomes evident that the Working Backwards Strategy beckons individuals and organizations to re-evaluate traditional paradigms in favor of a more outcome-centric and proactive approach to problem-solving. It is a call to action, urging us to adopt this strategy with a view toward fostering innovation and strategic growth in our respective endeavors.

Now is the moment to harness the transformative power of the Working Backwards Strategy . Embrace this strategic approach in your professional pursuits, educational goals, or personal development plans. Tackle your current challenges with an innovative lens and backward vision. For those seeking to delve deeper into the intricacies of strategic problem-solving, resources abound; extending from literature to problem solving courses and online MBA courses designed to inculcate the principles of this efficacious strategy.

Equip yourself with the knowledge and tools necessary to convert your aspirations into achievements. Remember, the journey to success is often charted by looking to the end and crafting your pathway backward.

## What is the conceptual framework of the working backwards approach to problem-solving?

The working backwards approach: a conceptual framework.

The working backwards approach stands as a revered method in problem-solving. It starts with the desired outcome. This involves envisioning the ideal solution or end-point before anything else. After defining the end goal, one must then deconstruct it, breaking down the steps that lead to this outcome, in reverse. This retrospective path mapping distinguishes the working backwards strategy from other techniques.

## Core Principles of Working Backwards

Identify the end goal first . Clarity of the objective remains paramount. Without a clear target, the process loses direction.

Deconstruct the problem . Break it down into smaller, manageable segments. Each segment acts as a milestone. Experts agree on the efficacy of this subdivision.

Reverse engineer the process . Working from the end backwards, the solver identifies the prerequisites for each milestone. This allows for a clear roadmap of actions.

Avoid assumptions . Each step must rest on solid, verifiable facts. False assumptions can derail the entire process.

Iterative refinement . Solutions often require tweaking. With each iteration, the solution becomes more precise.

## Advantages of Working Backwards

- Enhanced clarity . Visualizing the end state brings focus.

- Reduction of complexity . Breaking down the problem simplifies it.

- Strategic insight . Recognizing dependencies becomes straightforward.

- Encourages creativity . Solutions may emerge in counter-intuitive ways.

## Implementation in Different Fields

In software development, Amazon pioneered this approach. They write the press release before product development. Educators also apply this approach. They define learning outcomes before creating teaching plans. In research, scientists determine the desired discovery, then design experiments accordingly.

## Challenges and Considerations

Resource constraints may limit the applicability. Adjusting the goal can sometimes be necessary. Solvers must remain cognizant of potential roadblocks . They might encounter stages where progression is not evident in reverse.

## Theoretical Underpinnings

Cognitive science supports the working backwards method. It aligns with how the human brain problem-solves. The temporal structure of human cognition suggests a natural affinity for retrospective organization. Thus, working backwards proves not just effective, but intuitive.

The working backwards strategy offers a structured path to problem-solving. Begin with the end. Deconstruct to comprehend. Engage with facts. Refine rigorously. Recognize its potential across disciplines but remain aware of challenges. Anchored in cognitive theory, it mirrors our inherent reasoning processes. In sum, it presents a robust framework for tackling complex problems.

## How can the concept of working backwards enhance strategic decision-making in challenging situations?

Working backwards to enhance decision-making.

Strategic decision-making often involves complex problems. These problems can overwhelm decision-makers. When faced with such challenges, working backwards can clarify objectives and illuminate the path to achieving them.

## Begin with the End in Mind

The mantra "begin with the end in mind" is central here. It compels leaders to envision the desired outcome first. This future state acts as a North Star, guiding all subsequent decisions and actions.

## Set Clear Objectives

It is crucial to set clear, measurable objectives. These targets help assess decisions. They act as benchmarks throughout the process. Decision-makers can ensure alignment with the final goal by continually referencing these objectives.

## Map the Steps Reversely

Mapping steps from the end goal to the present is a reverse chronology. This action breaks down the path to success into manageable sections. Each step becomes smaller, more tangible. It simplifies the process of identifying what actions to take and when.

## Identify Key Milestones

Key milestones mark progress. They provide structure to the journey. Recognizing these milestones allows teams to celebrate small wins. This boosts morale and keeps focus sharp.

## Anticipate Potential Obstacles

Working backwards forces foresight of obstacles. Anticipated issues can be tackled proactively. Solutions become embedded in the plan from the start.

## Allocate Resources Wisely

Knowing the end goal helps allocate resources wisely. It becomes clear where to invest time, money, and effort. Scarce resources are conserved for critical paths.

## Adjust with Flexibility

Even the best plans need adjustments. With a clear end goal, decision-makers can pivot with confidence. They remain flexible without losing sight of the objective.

## Foster a Strategic Mindset

The backward approach fosters a strategic mindset. This thought process values proactive planning. It involves considering long-term impact over short-term gains.

## Enhance Team Communication

Clear benchmarks improve team communication. Everyone understands the direction and their role in it. This alignment reduces misunderstandings and enhances collective performance.

## Encourage Analytical Thinking

Working backwards encourages analytical thinking. Decision-makers must dissect the goal into its parts. They learn to understand the mechanisms that drive success.

Working backwards is not just a tactic. It is a strategic philosophy. It simplifies complex decision-making. Clarity, foresight, and structured planning are its byproducts. In challenging situations, this method can be a game-changer. Strategic leadership demands such innovative approaches. It stands as a testament to the virtue of thoughtful, reverse-engineered problem-solving.

## Can you provide an example of how employing a "working backwards" strategy has been successful in a real life complex challenge?

The virtue of working backwards.

Success often stems from innovative strategy. Employing 'working backwards' illustrates this point. This approach demands starting from the end goal. One then maps out the path to that goal. This ensures efficiency and focus.

Consider Amazon's approach to product development. They start with customer needs. They create the press release first. This defines the desired outcome. They write about the product's release. This draft press release centers on the customer experience. It covers benefits and customer challenges. Amazon then engineers the product backwards. They develop features to fulfill the press release claims. This ensures the final product delights customers.

## Real-Life Success Stories

This strategy thrives across various industries. Look at software development, for instance. Successful teams don't start coding immediately. They first define the user's end experience. Features then get planned. Every feature must serve the user's needs. Only after meticulous planning does coding begin.

Apple also embraces this mindset. They begin with the user experience. Product design comes next. Engineering follows to meet these specifications.

## The 'Working Backwards' Framework

This strategy can be broken down into steps:

- Identify the ultimate goal.

- Craft the final outcome's narrative.

- Pinpoint essential success metrics.

- Develop milestones working backwards from the goal.

- Execute the plan, ensuring alignment at each step.

## Critical Evaluation

While powerful, 'working backwards' is not infallible. It requires deep customer understanding. It demands that the end goal remains clear. Teams must adopt a customer-centric vision. Assumptions need constant testing. The strategy thrives on flexibility.

In summary, 'working backwards' breeds innovation. It demands a customer-first approach. It enforces clarity and focus throughout a project. Both Amazon and Apple showcase its success. Adopting this tactic means aligning with customer value. It ultimately leads to products that resonate in the market.

He is a content producer who specializes in blog content. He has a master's degree in business administration and he lives in the Netherlands.

## 5 Whys: Analyzing Root Causes for Effective Decision Making

## Mastering Problem Solving Skills: How Thinking Aloud Works

## Design Sprint 2.0: Streamline Innovative Solutions

## Unlock Problem Solving: Activeness is Key

- 7.3 Problem Solving
- Introduction
- 1.1 What Is Psychology?
- 1.2 History of Psychology
- 1.3 Contemporary Psychology
- 1.4 Careers in Psychology
- Review Questions
- Critical Thinking Questions
- Personal Application Questions
- 2.1 Why Is Research Important?
- 2.2 Approaches to Research
- 2.3 Analyzing Findings
- 3.1 Human Genetics
- 3.2 Cells of the Nervous System
- 3.3 Parts of the Nervous System
- 3.4 The Brain and Spinal Cord
- 3.5 The Endocrine System
- 4.1 What Is Consciousness?
- 4.2 Sleep and Why We Sleep
- 4.3 Stages of Sleep
- 4.4 Sleep Problems and Disorders
- 4.5 Substance Use and Abuse
- 4.6 Other States of Consciousness
- 5.1 Sensation versus Perception
- 5.2 Waves and Wavelengths
- 5.4 Hearing
- 5.5 The Other Senses
- 5.6 Gestalt Principles of Perception
- 6.1 What Is Learning?
- 6.2 Classical Conditioning
- 6.3 Operant Conditioning
- 6.4 Observational Learning (Modeling)
- 7.1 What Is Cognition?
- 7.2 Language
- 7.4 What Are Intelligence and Creativity?
- 7.5 Measures of Intelligence
- 7.6 The Source of Intelligence
- 8.1 How Memory Functions
- 8.2 Parts of the Brain Involved with Memory
- 8.3 Problems with Memory
- 8.4 Ways to Enhance Memory
- 9.1 What Is Lifespan Development?
- 9.2 Lifespan Theories
- 9.3 Stages of Development
- 9.4 Death and Dying
- 10.1 Motivation
- 10.2 Hunger and Eating
- 10.3 Sexual Behavior, Sexuality, and Gender Identity
- 10.4 Emotion
- 11.1 What Is Personality?
- 11.2 Freud and the Psychodynamic Perspective
- 11.3 Neo-Freudians: Adler, Erikson, Jung, and Horney
- 11.4 Learning Approaches
- 11.5 Humanistic Approaches
- 11.6 Biological Approaches
- 11.7 Trait Theorists
- 11.8 Cultural Understandings of Personality
- 11.9 Personality Assessment
- 12.1 What Is Social Psychology?
- 12.2 Self-presentation
- 12.3 Attitudes and Persuasion
- 12.4 Conformity, Compliance, and Obedience
- 12.5 Prejudice and Discrimination
- 12.6 Aggression
- 12.7 Prosocial Behavior
- 13.1 What Is Industrial and Organizational Psychology?
- 13.2 Industrial Psychology: Selecting and Evaluating Employees
- 13.3 Organizational Psychology: The Social Dimension of Work
- 13.4 Human Factors Psychology and Workplace Design
- 14.1 What Is Stress?
- 14.2 Stressors
- 14.3 Stress and Illness
- 14.4 Regulation of Stress
- 14.5 The Pursuit of Happiness
- 15.1 What Are Psychological Disorders?
- 15.2 Diagnosing and Classifying Psychological Disorders
- 15.3 Perspectives on Psychological Disorders
- 15.4 Anxiety Disorders
- 15.5 Obsessive-Compulsive and Related Disorders
- 15.6 Posttraumatic Stress Disorder
- 15.7 Mood and Related Disorders
- 15.8 Schizophrenia
- 15.9 Dissociative Disorders
- 15.10 Disorders in Childhood
- 15.11 Personality Disorders
- 16.1 Mental Health Treatment: Past and Present
- 16.2 Types of Treatment
- 16.3 Treatment Modalities
- 16.4 Substance-Related and Addictive Disorders: A Special Case
- 16.5 The Sociocultural Model and Therapy Utilization

## Learning Objectives

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

- Describe problem solving strategies
- Define algorithm and heuristic
- Explain some common roadblocks to effective problem solving and decision making

People face problems every day—usually, multiple problems throughout the day. Sometimes these problems are straightforward: To double a recipe for pizza dough, for example, all that is required is that each ingredient in the recipe be doubled. Sometimes, however, the problems we encounter are more complex. For example, say you have a work deadline, and you must mail a printed copy of a report to your supervisor by the end of the business day. The report is time-sensitive and must be sent overnight. You finished the report last night, but your printer will not work today. What should you do? First, you need to identify the problem and then apply a strategy for solving the problem.

## Problem-Solving Strategies

When you are presented with a problem—whether it is a complex mathematical problem or a broken printer, how do you solve it? Before finding a solution to the problem, the problem must first be clearly identified. After that, one of many problem solving strategies can be applied, hopefully resulting in a solution.

A problem-solving strategy is a plan of action used to find a solution. Different strategies have different action plans associated with them ( Table 7.2 ). For example, a well-known strategy is trial and error . The old adage, “If at first you don’t succeed, try, try again” describes trial and error. In terms of your broken printer, you could try checking the ink levels, and if that doesn’t work, you could check to make sure the paper tray isn’t jammed. Or maybe the printer isn’t actually connected to your laptop. When using trial and error, you would continue to try different solutions until you solved your problem. Although trial and error is not typically one of the most time-efficient strategies, it is a commonly used one.

Another type of strategy is an algorithm. An algorithm is a problem-solving formula that provides you with step-by-step instructions used to achieve a desired outcome (Kahneman, 2011). You can think of an algorithm as a recipe with highly detailed instructions that produce the same result every time they are performed. Algorithms are used frequently in our everyday lives, especially in computer science. When you run a search on the Internet, search engines like Google use algorithms to decide which entries will appear first in your list of results. Facebook also uses algorithms to decide which posts to display on your newsfeed. Can you identify other situations in which algorithms are used?

A heuristic is another type of problem solving strategy. While an algorithm must be followed exactly to produce a correct result, a heuristic is a general problem-solving framework (Tversky & Kahneman, 1974). You can think of these as mental shortcuts that are used to solve problems. A “rule of thumb” is an example of a heuristic. Such a rule saves the person time and energy when making a decision, but despite its time-saving characteristics, it is not always the best method for making a rational decision. Different types of heuristics are used in different types of situations, but the impulse to use a heuristic occurs when one of five conditions is met (Pratkanis, 1989):

- When one is faced with too much information
- When the time to make a decision is limited
- When the decision to be made is unimportant
- When there is access to very little information to use in making the decision
- When an appropriate heuristic happens to come to mind in the same moment

Working backwards is a useful heuristic in which you begin solving the problem by focusing on the end result. Consider this example: You live in Washington, D.C. and have been invited to a wedding at 4 PM on Saturday in Philadelphia. Knowing that Interstate 95 tends to back up any day of the week, you need to plan your route and time your departure accordingly. If you want to be at the wedding service by 3:30 PM, and it takes 2.5 hours to get to Philadelphia without traffic, what time should you leave your house? You use the working backwards heuristic to plan the events of your day on a regular basis, probably without even thinking about it.

Another useful heuristic is the practice of accomplishing a large goal or task by breaking it into a series of smaller steps. Students often use this common method to complete a large research project or long essay for school. For example, students typically brainstorm, develop a thesis or main topic, research the chosen topic, organize their information into an outline, write a rough draft, revise and edit the rough draft, develop a final draft, organize the references list, and proofread their work before turning in the project. The large task becomes less overwhelming when it is broken down into a series of small steps.

## Everyday Connection

Solving puzzles.

Problem-solving abilities can improve with practice. Many people challenge themselves every day with puzzles and other mental exercises to sharpen their problem-solving skills. Sudoku puzzles appear daily in most newspapers. Typically, a sudoku puzzle is a 9×9 grid. The simple sudoku below ( Figure 7.7 ) is a 4×4 grid. To solve the puzzle, fill in the empty boxes with a single digit: 1, 2, 3, or 4. Here are the rules: The numbers must total 10 in each bolded box, each row, and each column; however, each digit can only appear once in a bolded box, row, and column. Time yourself as you solve this puzzle and compare your time with a classmate.

Here is another popular type of puzzle ( Figure 7.8 ) that challenges your spatial reasoning skills. Connect all nine dots with four connecting straight lines without lifting your pencil from the paper:

Take a look at the “Puzzling Scales” logic puzzle below ( Figure 7.9 ). Sam Loyd, a well-known puzzle master, created and refined countless puzzles throughout his lifetime (Cyclopedia of Puzzles, n.d.).

## Pitfalls to Problem Solving

Not all problems are successfully solved, however. What challenges stop us from successfully solving a problem? Imagine a person in a room that has four doorways. One doorway that has always been open in the past is now locked. The person, accustomed to exiting the room by that particular doorway, keeps trying to get out through the same doorway even though the other three doorways are open. The person is stuck—but they just need to go to another doorway, instead of trying to get out through the locked doorway. A mental set is where you persist in approaching a problem in a way that has worked in the past but is clearly not working now.

Functional fixedness is a type of mental set where you cannot perceive an object being used for something other than what it was designed for. Duncker (1945) conducted foundational research on functional fixedness. He created an experiment in which participants were given a candle, a book of matches, and a box of thumbtacks. They were instructed to use those items to attach the candle to the wall so that it did not drip wax onto the table below. Participants had to use functional fixedness to overcome the problem ( Figure 7.10 ). During the Apollo 13 mission to the moon, NASA engineers at Mission Control had to overcome functional fixedness to save the lives of the astronauts aboard the spacecraft. An explosion in a module of the spacecraft damaged multiple systems. The astronauts were in danger of being poisoned by rising levels of carbon dioxide because of problems with the carbon dioxide filters. The engineers found a way for the astronauts to use spare plastic bags, tape, and air hoses to create a makeshift air filter, which saved the lives of the astronauts.

## Link to Learning

Check out this Apollo 13 scene about NASA engineers overcoming functional fixedness to learn more.

Researchers have investigated whether functional fixedness is affected by culture. In one experiment, individuals from the Shuar group in Ecuador were asked to use an object for a purpose other than that for which the object was originally intended. For example, the participants were told a story about a bear and a rabbit that were separated by a river and asked to select among various objects, including a spoon, a cup, erasers, and so on, to help the animals. The spoon was the only object long enough to span the imaginary river, but if the spoon was presented in a way that reflected its normal usage, it took participants longer to choose the spoon to solve the problem. (German & Barrett, 2005). The researchers wanted to know if exposure to highly specialized tools, as occurs with individuals in industrialized nations, affects their ability to transcend functional fixedness. It was determined that functional fixedness is experienced in both industrialized and nonindustrialized cultures (German & Barrett, 2005).

In order to make good decisions, we use our knowledge and our reasoning. Often, this knowledge and reasoning is sound and solid. Sometimes, however, we are swayed by biases or by others manipulating a situation. For example, let’s say you and three friends wanted to rent a house and had a combined target budget of $1,600. The realtor shows you only very run-down houses for $1,600 and then shows you a very nice house for $2,000. Might you ask each person to pay more in rent to get the $2,000 home? Why would the realtor show you the run-down houses and the nice house? The realtor may be challenging your anchoring bias. An anchoring bias occurs when you focus on one piece of information when making a decision or solving a problem. In this case, you’re so focused on the amount of money you are willing to spend that you may not recognize what kinds of houses are available at that price point.

The confirmation bias is the tendency to focus on information that confirms your existing beliefs. For example, if you think that your professor is not very nice, you notice all of the instances of rude behavior exhibited by the professor while ignoring the countless pleasant interactions he is involved in on a daily basis. Hindsight bias leads you to believe that the event you just experienced was predictable, even though it really wasn’t. In other words, you knew all along that things would turn out the way they did. Representative bias describes a faulty way of thinking, in which you unintentionally stereotype someone or something; for example, you may assume that your professors spend their free time reading books and engaging in intellectual conversation, because the idea of them spending their time playing volleyball or visiting an amusement park does not fit in with your stereotypes of professors.

Finally, the availability heuristic is a heuristic in which you make a decision based on an example, information, or recent experience that is that readily available to you, even though it may not be the best example to inform your decision . Biases tend to “preserve that which is already established—to maintain our preexisting knowledge, beliefs, attitudes, and hypotheses” (Aronson, 1995; Kahneman, 2011). These biases are summarized in Table 7.3 .

Watch this teacher-made music video about cognitive biases to learn more.

Were you able to determine how many marbles are needed to balance the scales in Figure 7.9 ? You need nine. Were you able to solve the problems in Figure 7.7 and Figure 7.8 ? Here are the answers ( Figure 7.11 ).

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- Book URL: https://openstax.org/books/psychology-2e/pages/1-introduction
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## Primary 3 Working Backwards & Its Method

Math heuristics for problem solving, primary 3 working backwards & its method, what is working backwards in math.

The scenario occurs when the quantity data is insufficient to work from the beginning . Working Backwards is a problem-solving strategy in which you start with the end goal and work backward to figure out the steps needed to get there. In other words, instead of starting from the beginning and moving forward, you start from the end and move backward. This strategy is commonly used in math problems that ask you to find a starting value or figure out what happened before a given situation.

## How to Solve Math Questions with Working Backwords Method?

Let's take a look at this primary 3 word problem example:.

Sarah had some pens. She bought 34 pens. She then threw away 29 pens as they were spoilt. In the end, she had 64 pens. How many pens did Sarah have at first?

Identify the Concept

We know this is a Working Backwards question as…

Workings Explained

Always remember when we work backwards, everything will be reversed. Example the 2nd sentence – “She bought 34 pens”. We know when we buy things, we will have more. We need to add. However, when we work backwards, instead of adding, we need to subtract.

- We will start drawing the model from the end by drawing a box and label it “End”. Put the end amount “64” in the box.
- Draw arrow to point to the left, draw another box. On top of the arrow write “+29” as “Sarah threw away 29 pens”. Instead of subtract, we need to add. In the box, write “93” (64+29=93).
- Draw another arrow to point to the left, draw another box. On top of the arrow, write “-34” as Sarah bought 34 pens. Instead of adding, we need to subtract. In the box, write “59” (93-34=59). Label the box “At First” or “Before”.

Sarah had 59 pens at first.

We know this is a Working Backwards question as we were not told the number of pens Sarah had but were asked for the number of pens she had at first.

Or search by topic

## Number and algebra

- The Number System and Place Value
- Calculations and Numerical Methods
- Fractions, Decimals, Percentages, Ratio and Proportion
- Properties of Numbers
- Patterns, Sequences and Structure
- Algebraic expressions, equations and formulae
- Coordinates, Functions and Graphs

## Geometry and measure

- Angles, Polygons, and Geometrical Proof
- 3D Geometry, Shape and Space
- Measuring and calculating with units
- Transformations and constructions
- Pythagoras and Trigonometry
- Vectors and Matrices

## Probability and statistics

- Handling, Processing and Representing Data
- Probability

## Working mathematically

- Thinking mathematically
- Mathematical mindsets
- Cross-curricular contexts
- Physical and digital manipulatives

## For younger learners

- Early Years Foundation Stage

## Advanced mathematics

- Decision Mathematics and Combinatorics
- Advanced Probability and Statistics

## Working Backwards at KS2

Working backwards can be a very useful problem-solving skill. These activities lend themselves to being tackled in this way.

This collection is one of our Primary Curriculum collections - tasks that are grouped by topic.

Can you work out how to win this game of Nim? Does it matter if you go first or second?

## Junior Frogs

Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?

In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

## Would You Rather?

Would you rather: Have 10% of Â£5 or 75% of 80p? Be given 60% of 2 pizzas or 26% of 5 pizzas?

## First Connect Three

Add or subtract the two numbers on the spinners and try to complete a row of three. Are there some numbers that are good to aim for?

## Multiplication Squares

Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.

## What's in the Box?

This big box multiplies anything that goes inside it by the same number. If you know the numbers that come out, what multiplication might be going on in the box?

## Andy's Marbles

Andy had a big bag of marbles but unfortunately the bottom of it split and all the marbles spilled out. Use the information to find out how many there were in the bag originally.

Can you go through this maze so that the numbers you pass add to exactly 100?

## All the Digits Live

This multiplication uses each of the digits 0 - 9 once and once only. Using the information given, can you replace the stars in the calculation with figures?

## Mystery Matrix

Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.

## Counting Cards

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

## Nice or Nasty

There are nasty versions of this dice game but we'll start with the nice ones...

## Missing Multipliers

What is the smallest number of answers you need to reveal in order to work out the missing headers?

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

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## Share Podcast

## Stuck on a Problem? Try Switching Up Your Approach

A conversation with Cheryl Strauss Einhorn on decision-making dexterity.

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Many leaders confidently go about tackling challenges. After all, relying on their experience got them to where they are. But taking the same approach over and over again can actually hold you back. Sometimes you need to switch up your tactics to break through to the next level. Decision-making expert Cheryl Strauss Einhorn says the first step is to understand your personal problem-solving style. Then she explains a framework to assess the situation and select the best approach. Einhorn is founder and CEO of Decisive. She also wrote the book Problem Solver: Maximizing Your Strengths to Make Better Decisions and the HBR article “ When Your Go-To Problem-Solving Approach Fails .”

CURT NICKISCH: Welcome to the HBR IdeaCast from Harvard Business Review. I’m Curt Nickisch.

At this stage in your career, whether you realize it or not, you probably lean on the same framework to make decisions. Call it a habit or a pattern, whether it’s unconscious or deliberate. Like it or not, you have developed your own tried and true decision-making style. But is it really the best way?

Often the model that you turn to to solve problems and make decisions is the one you’ve grown comfortable with, but it’s not necessarily the most effective one for that situation. Even the best leaders sometimes need a refresh, and that’s especially true when your default doesn’t seem to be working in a new scenario.

Our guest today studies the behaviors and psychology behind making decisions. In fact, she has identified five different archetypes. She says that the key to solving the latest challenge you face might be in understanding your own style and knowing when to switch up your approach.

Cheryl Strauss Einhorn is the founder and CEO of Decisive, a decision sciences consultancy. She also wrote the book Problem Solver and the HBR article “When Your Go-To Problem Solving Approach Fails.” Cheryl, thanks for joining.

CHERYL STRAUSS EINHORN: Thank you so much for having me.

CURT NICKISCH: So we’re going to talk about the main kinds of decision makers that you’ve identified in a bit. But I want to start by asking, why do we tend to fall back on certain patterns or behaviors around decision making? Where does that come from?

CHERYL STRAUSS EINHORN: Well, there’s really two reasons. First, we have comfortable ways of being. If you think about it, most of us are as people who are right-handed, more comfortable doing things with our right hand. Or if we’re left-handed, left-hand. And our dominant ways of being as decision-makers are similar. It’s a comfortable way that we work in the world. And then the second point is that the way that we present ourselves, those habits and patterns that are comfortable, they speak to an underlying set of values, and those are the values that underpin how we make our decisions.

CURT NICKISCH: And is this any different for business leaders that are thrust into new situations, it’s almost like maybe you’re forced to use your left hand, where normally you wouldn’t?

CHERYL STRAUSS EINHORN: I think for somebody who’s very experienced, they have habits and patterns of behaving that have really worked well for them. And I think they then therefore may tend to have a stronger bias to do those things that have worked well. And so it is in really trying to pry open that cognitive space to allow for new information and new thinking that can really give them the opportunity therefore to do something different.

CURT NICKISCH: So it almost sounds like it might actually be harder for a business leader because they’re thinking of themselves, “The way I do things has been successful. It’s gotten me to this point, and because of that, you might have a little bit of a failure of imagination.

CHERYL STRAUSS EINHORN: And also a real comfort that you have this data set behind you of success. And so I think that that’s right.

CURT NICKISCH: So before we get into trying to solve a problem when your standard approach just isn’t working, we should talk about the different kinds of problem solvers that you’ve identified through your work and research: these are adventurers, detectives, listeners, thinkers, and visionaries. Can you quickly run through each of those and just give us the main attributes?

CHERYL STRAUSS EINHORN: So I gave them each fun names because we think in language. The adventurer is somebody who makes decisions easily, they tend to go with their gut, but therefore they may downplay the evidence and input from others, especially if it contradicts their gut reaction.

For detectives, and I’m a detective, we’re people who like to follow the data. A downside of that may be that we overvalue facts and we undervalue people.

Listeners are our most collaborative and cooperative decision makers. They generally want to solicit the input of others, but sometimes they have difficulty accessing their own inner voice.

Thinkers are people who have the action in between their ears. They want to understand the why. They thrive on identifying multiple pathways and outcome, but they can struggle to make decision in a timely manner as they tend to end up in a frame blindness looking at one option against the other, which may miss the bigger picture.

And then we have our visionaries. They pride themselves on seeing pathways that others don’t, but therefore they may have a scarcity bias or want to avoid the ordinary even when it can be effective. And so I hope what you can see here is that each of these approaches brings a different underlying value structure and therefore they’re optimizing for different things in the way that they solve problems.

CURT NICKISCH: I just thought to myself what style I probably am. And I bet a lot of listeners did the same thing. Let me give you an example of something you might have to do as a manager: catering for a business meeting. You’ve got a bunch of people in, maybe people from different teams, not your normal meeting, and you have to decide how to feed everybody for lunch during this long meeting. How would each of those archetypes or personality styles choose that differently?

CHERYL STRAUSS EINHORN: Well, I think that this is something that many people can relate to, right? So in the catering example, the adventurer might look at the menu of things that she could be offering and she’s going to immediately pick one and say, “That looks good.” Because she’s going to feel a natural instinct and she can get the decision over pretty quickly because she feels confident.

The detective is going to look at the different ingredients and think about what would be a really good option that everybody might be able to eat and she’s going to be anchored in that detail for instance, “Oh, I can see that this would be something that would be acceptable to a lot of people,” and she’ll make the choice based on those facts.

For the listener, she might really want to be taking into account the different eating needs that everybody has and how hot the room might be. And she might be thinking about what would be comfortable if it had to sit out, for example, and how to really make sure that everybody feels welcome in that moment because focused on the people.

CURT NICKISCH: Does she actually go out and ask people what they want or do a poll?

CHERYL STRAUSS EINHORN: Absolutely. She might go out and ask people. She might say, “Hey, I’d like to ask everybody, ‘Are there any food allergies?’.” Because she absolutely is focused on making sure that the people feel included and that she’s doing this in a collaborative, cooperative way.

For the thinker, she may look at the different options and weigh one against another. “This meal might make everybody feel really full, but this meal might be more well-balanced and so on,” and really spend time thinking about how can she mitigate the downside because the thinker has a loss aversion. They would rather make sure that the decision doesn’t fail than optimize for the best possible outcome or the best outcome possible.

And then the visionary might look at the choices and say, “Well, I like this dish, but I like the sauce from something else,” and might ask if she can create something that’s not even exactly on the menu.

CURT NICKISCH: So that’s kind of fascinating, and I think that kind of helps each of us listening sort of even better identify what type we might have. When does this normally come up where you realize that your style doesn’t always work?

CHERYL STRAUSS EINHORN: Well, there are times where we have something that I call situationality that can get in the way. Situationality is the culmination of many factors within context. It includes our location, our life stage, our decision ownership, and our team dynamics, and sometimes that can be a very good thing.

So if you’re an adventurer for example, and you normally have a gut inclination that tells you how you’d like to proceed, if you’re in a brand new environment, maybe you’re starting a new job, to show up as a listener and take time to hear maybe the pathway that this organization wants you to follow that can help you to build a relationship and to also take a temperature, “Was my gut inclination aligned with how this organization actually wants to make this decision?”

CURT NICKISCH: The key lesson there is that that style got you to get that job, but it doesn’t always mean that that style is going to be the most successful in that job.

CHERYL STRAUSS EINHORN: Or it might not be successful right at the outset when you’re just initially meeting the people and learning about the job responsibilities itself for that particular organization. I think that’s exactly right. At other times situationality can get in our way. We might have a dominant way of being, and then something happens and all of a sudden we end up in a situation with a lot of friction.

And I’ll give you an example. I was working with a visionary CEO of a geospatial technology company. As a visionary, he really is focused on this big picture vision of the good that he can bring to the world with his new technology, and it gives him a lot of flexibility for how to get to that vision.

But at one point, when he was accepted into a prestigious National Science Foundation program, he ended up hearing information that really caused him concern. He acted as a detective, shrinking his worldview to really focus and zero in on these details that he didn’t like and got so stuck in the weeds that he had a lot of problems with the leadership of the program.

In working together, we were able to examine that all of a sudden the detective had intruded in this situation to override his visionary. Once we were able to look at that, he was able to realign that in being the visionary and holding onto this picture of the good that he could be bringing to the world, he was better able to lift himself out of those details to get back on board that the leadership of this program was actually trying to help him all along to bring his beautiful vision to the world, and that decreased the friction and enabled him to succeed.

CURT NICKISCH: I definitely see a strong case here for changing your style depending on the situation. What do you do first as you realize you’re hitting your head against the wall on a problem or something that you can’t seem to figure out? What’s the first question that you ask yourself in that situation?

CHERYL STRAUSS EINHORN: I think the first thing you want to do is to define the problem. What’s the negative experience that you’re having in the workplace that requires a decision? And then ask yourself to assess the location and team dynamics. Where are you working? Are you in an office or remote? Are you part of a team? Are you independent? And so on.

And then from there, I think you can consider your own career stage. Are you starting? Are you at the peak? Might you be counting your days until retirement? And then you can think about the other people’s perspective that you’re working with. Who are those stakeholders? How are they trying to make the situation work or not work? And then you can look at your own level of decision ownership, right? Is it your decision to make or how much will you be impacted by the decision outcome?

And then you can connect this situationality back to your problem solver profile. And you can look at whether or not you’re showing up in one of the other profiles, and then you can return to look at your own problem solver profile because it’s usually returning back to what that dominant profile is that you can lean into the strengths of that approach to help you right the ship and make more effective decisions with others.

CURT NICKISCH: So what else can you do to try to overcome the blind spots or the patterns behavior that you typically fall back on?

CHERYL STRAUSS EINHORN: I think the first thing is to gain awareness. So for instance, for an adventurer, she has an optimism bias. She knows that if she makes a decision and it goes awry, guess what? The next decision is always ahead. And so she might tend to gloss over facts or important details. And so recognizing that there is that optimism bias can really be very useful to help you check and challenge it.

Similarly, for the detective who likes evidence, she’s often somebody who feels very comfortable doing research, and therefore she can have an underlying confirmation bias where she can go and identify how a specific piece of information favors the hypothesis that she has, but that’s not as good or as rigorous as looking for disconfirming data. And so again, knowing your problem solver profile, the strengths and the related cognitive biases is the first step to being able to really build your awareness so that you can have an opportunity to check and challenge what you normally do.

CURT NICKISCH: I thought it was interesting what you said earlier about somebody close to retirement, how that might change the decision they make or how they approach it. How else can this change depending on the stage of your career?

CHERYL STRAUSS EINHORN: Well, I think in this stage of your career, we were talking about before that you might have somebody who’s very experienced, so they have a way of thinking about how a decision should be made that provides them knowledge, but it also may mean that they have more biases associated with the way that it’s been done. If you’re a seasoned professional, you may be able to speak to a problem that you’ve solved before with expertise, and that might make it more difficult to reconsider how you approach the problem. That’s not always true, but certainly the experience content to narrow what we think the actual answer should be instead of expanding it.

CURT NICKISCH: Yeah, I wonder if sometimes you realize that the decision that you’re struggling with is actually not your decision, or maybe you have claimed too much of it, then you realize? How much does decision ownership play into this process?

CHERYL STRAUSS EINHORN: I think decision ownership is something that’s underappreciated, right? Thinking about, is it your decision to make? How much will you or your organization be impacted? And how important is the decision to you or your organization? Impact and importance are obviously not the same thing, because impact is the effect on someone or something and importance is the significance or the value. So you can have a decision that can have a significant impact, but be of little importance and vice versa.

So thinking about whether or not it’s your decision to actually make and how much it requires input and sign off from others can really help you to see whether or not you have worked well enough to include the voices of the other stakeholders. Because holistic problem solving occurs when you are actively including the other people who are going to be impacted by the decision that’s being made.

CURT NICKISCH: Is somebody who is a listener, are they just by default more successful in these situations? Is it basically just trying to change your style so that you are more of a listener? Or is decision ownership more than that?

CHERYL STRAUSS EINHORN: That’s a very interesting point you’re making, because for a listener, they would actively want to be including people to make a decision. For a detective, they may or may not include other people because they naturally want to go and find the data. For an adventurer, they’d want to make the decision pretty quickly because they have a pretty strong instinct on what that decision should be and so on. So each of the different problem solver profiles might view decision ownership differently?

CURT NICKISCH: Yeah. A listener could make the mistake of listening to the wrong people.

CHERYL STRAUSS EINHORN: Well, a listener usually has a trusted group of advisors, and therefore they can have an underlying liking bias where they overweight information that comes from people that they have an affinity for. That may make it more difficult for them to bring in outside voices beyond that. And it also may make it difficult at times for them to really identify their own inner voice.

CURT NICKISCH: Once you’ve learned your style, understand its shortcomings and have gotten better about switching up your approach to fit the scenario that you’re in, does this become a new habit or do you really have to keep working on it deliberately all the time as you go along?

CHERYL STRAUSS EINHORN: I think the problem solver profiles give you a beautiful opportunity to be a more active listener. If you recognize, for example, that your problem solver profile is one that may have friction with another problem solver profile, really listening for clues as to how somebody else is approaching the decision can help you use the skills of the other problem solver to work better together. And at the same time, once you really have this opportunity to learn all the profiles, you can actively try them on. If you are a detective, you could go to the supermarket as a visionary. Or you might take a vacation as an adventurer and so on. And so by using the different profiles that are not yours, you can have an opportunity to see where the discomfort is and then to try to work through that discomfort so that you can become more dynamic. And over time, it will become easier as you begin to lean into trying on the different profiles for yourself, in decisions where you feel comfortable stepping out of that natural habit and pattern of making decisions.

CURT NICKISCH: The other thing I’m kind of realizing is that part of the situationality is the organization that you’re in. We talk about organizations as being very data-driven or analytical, right? And it may actually favor certain kinds of decision making and undervalue other kinds. And so, seeing how you fit in that organization can really make a big difference.

CHERYL STRAUSS EINHORN: That can definitely make a big difference. I’ve been doing a program lately with the federal government. Each of these organizations, whether it’s the Treasury or the IRS and so on, they each have a real process for how things take place. So in one of our recent workshops, we had a discussion about where’s the place for the adventurer? How does this person actually fit into an organization like this? One of the things that we talked about is that the adventurer is such a nice person to have at the table because they really can help the trains to move on time and build a momentum.

The other thing that they can do, for example, in an organization that seems maybe slower, more aligned with thinkers and detectives, is that they also don’t get anchored on a particular pathway forward. They have a beautiful flexibility to be able to hear a lot of ideas, instinctively be able to switch between them and to identify why it is that they feel like a specific pathway forward might be the best way to go.

And so recognizing that the organization can seem to favor certain types of profiles and then recognizing how the intellectual diversity can still work to augment how the organization works, I think, is something that is really beautiful.

CURT NICKISCH: How does this factor when you’re working in a team? You have your own decision-making style, but if you’re working on a team that’s putting different styles together and has a different dynamic and I’m just curious how that plays out.

CHERYL STRAUSS EINHORN: So if you have a team that’s been together for a long time and you have a variety of profiles, let’s say that your CEO is an adventurer, but you have a team that’s on that senior leadership group that’s primarily thinkers, detectives, listeners, these have very different speeds of decision-making. So with a group like that, you may as the CEO, want to send out, “Gee, in advance, here’s an email framing the situation, the goal that we’re working on.” And it gives people a little bit of time to go at their own speed to do their own investigative process, the thinker to look at the options, the listener to make sure that they’ve canvassed, the stakeholders, the detectives, to gather their evidence.

So by the time they’re gathered around that conference room table, everybody’s actually ready for the decision-making instead of having to sit in the problem-solving. If you have a team that’s never worked together, you’re coming in the room, you don’t really know who people are in terms of their problem-solver profile, you might be able to ask people, “What is it that you need in order to make a decision?” And then you can hear, do they talk about facts? Do they talk about including stakeholders? Do they talk about understanding the options at the table?

Again, that active listening can help you then go into a situation with somebody who you’ve never worked with before, being able to better sync up on how your different problem-solver profiles can work well together.

CURT NICKISCH: Cheryl, you coach individuals, you also work in organizations. I’m just curious what the biggest misunderstanding is that people have about making decisions that you think can be corrected or cleared up?

CHERYL STRAUSS EINHORN: I think one of the biggest things is people don’t understand why there’s friction between people when solving problems, and they sort of throw their hands up initially and they can think about, “I’m uncomfortable with this personality.”

Well, decision-making is actually a part of personality that by knowing the problem-solver profiles can help you unblock your relationships to make better decisions together. As soon as people recognize themselves in the profiles, they have this light bulb go on, “This is why I’m behaving this way. This is what I value in my decisions.”

And it’s not personal that in understanding that your way is just one of five ways, it gives you an opening for how to understand what is somebody else’s incentive structure, what is their motivation for why they’re approaching the decision the way that they are. And that gives you a way to find this intersection of how to work well with them so that you can strengthen that relationship and make better decisions together.

CURT NICKISCH: Well, Cheryl, I think you’ve given listeners a little better sense of who they are and why they approach decisions and problems the way they do. Thanks for coming on the show to talk about it.

CHERYL STRAUSS EINHORN: Thank you so much for having me today.

CURT NICKISCH: That’s Cheryl Strauss Einhorn, founder and CEO of Decisive. She wrote the book Problem Solver and the HBR article When Your Go-To Problem Solving Approach Fails.

And we have nearly 1,000 episodes, plus more podcasts to help you manage your team, your organization, and your career, find them at HBR.org/podcasts or search HBR in Apple Podcasts, Spotify, or wherever you listen.

Thanks to our team, Senior Producer Mary Dooe, Associate Producer Hannah Bates, Audio Product Manager Ian Fox, and Senior Production Specialist Rob Eckhardt. Thank you for listening to the HBR IdeaCast . We’ll be back with a new episode on Tuesday. I’m Curt Nickisch.

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This article is about decision making and problem solving.

- Personal strategy and style
- Managing yourself
- Leadership styles
- Leading teams
- Cognitive bias

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Working backwards is to start with the final solution and work back one step at a time to get to the beginning. It may also be helpful for students to understand that this is useful in many aspects of life, not just solving math problems.

· Mar 18, 2023 -- Working backward is a problem-solving strategy where you start from the desired end result or goal and work your way back to the initial conditions or starting point....

The first step in working backwards is to start with the final number. The next step is to undo the problem step-by-step in chronological order. The end is when the beginning is determined. How...

Made famous by Amazon, the working backwards strategy is a favorite among many product teams and thought leaders. A press release is usually the very last step in the product development and launch process. It tells the world: "Here I am, this is what I can do, and this is why you should care."

This lesson will expand your toolbox of problem-solving strategies to include guess and check and working backward. Let's begin by reviewing the four-step problem-solving plan: Step 1: Understand the problem. Step 2: Devise a plan - Translate. Step 3: Carry out the plan - Solve. Step 4: Look - Check and Interpret.

Elementary Math Problem Solving - Working BackwardsIn this video, we explore one of eight problem-solving strategies for the primary math student. Students a...

Educator Maurice Ashley Imagine where you want to be someday. Now, how did you get there? Retrograde analysis is a style of problem solving where you work backwards from the endgame you want. It can help you win at chess -- or solve a problem in real life. At TEDYouth 2012, chess grandmaster Maurice Ashley delves into his

A problem-solving strategy is a plan of action used to find a solution. Different strategies have different action plans associated with them (table below). For example, a well-known strategy is trial and error. The old adage, "If at first you don't succeed, try, try again" describes trial and error.

Learn how to solve a problem by working backwards.We hope you are enjoying this video! For more in-depth learning, check out Miacademy.co (https://www.parent...

Abstract: The very name of this strategy sounds confusing to most people. It is a very unnatural way of doing things. When most of us went to school, we were taught to solve mathematical problems in a direct, straightforward manner. And yet, working backwards is the way many real-life problems are often resolved.

The Working Backwards Strategy is a beacon of hope amidst the formidable waves of complexity that confront individuals and organizations alike. In essence, this strategic approach is a clarion call to begin where most end - the desired outcome.

What should you do? First, you need to identify the problem and then apply a strategy for solving the problem. Problem-Solving Strategies When you are presented with a problem—whether it is a complex mathematical problem or a broken printer, how do you solve it? Before finding a solution to the problem, the problem must first be clearly identified.

Working Backwards is a problem-solving strategy in which you start with the end goal and work backward to figure out the steps needed to get there. In other words, instead of starting from the beginning and moving forward, you start from the end and move backward.

Among the range of problem solving strategies in mathematics, working backwards is a particularly useful method in situations when the end result of a problem is known and one has to find the initial quantity.

Problem Solving Strategy 3 (Using a variable to find the sum of a sequence.) Gauss's strategy for sequences. last term = fixed number (n-1) + first term. ... Problem Solving Strategy 4 (Working Backwards) This is considered a strategy in many schools. If you are given an answer, and the steps that were taken to arrive at that answer, you should ...

When introducing the different types of problem solving strategies; looking for patterns, make a list, draw a picture, guess and check, and working backwards, working backwards can sometimes be a fun one to start first. Giving the students a trick about reading through all the directions is unintimidating and proves a point in an amusing way.

Working backward Means-end analysis Recap Got a problem you're trying to solve? Strategies like trial and error, gut instincts, and "working backward" can help. We look at some examples...

In the working-backward approach, the problem solver starts at the end and works toward the beginning. For example, suppose one is planning a trip from New York City to Paris. One wishes to arrive at one's Parisian hotel. To arrive, one needs to take a taxi from… Home Health & Medicine Psychology & Mental Health Science & Tech heuristic psychology

Reversible Reasoning and the Working Backwards Problem Solving Strategy. Making sense of mathematical concepts and solving mathematical problems may demand different forms of reasoning. These could be either domain-based, such as algebraic, geometric or statistical reasoning, while others are more general such as inductive/deductive reasoning ...

Two players take turns at removing 1 to 4 coins from an original pile of 16 coins. Is there a winning strategy for either player? These short videos are mean...

Working Backwards at KS2. Working backwards can be a very useful problem-solving skill. These activities lend themselves to being tackled in this way. This collection is one of our Primary Curriculum collections - tasks that are grouped by topic.

problem solving tasks, and one of these is proof construction.... in sentential logic using an inference-replacement rule set. This empirical research identifies several strategies which facilitate problem solving, such as means-ends reasoning, difference reduction, and working backwards from goal toward starting state. As detailed below, these

Decision-making expert Cheryl Strauss Einhorn says the first step is to understand your personal problem-solving style. Then she explains a framework to assess the situation and select the best ...

The findings of the study showed that (1) the mathematical reasoning ability of the students who used and those who did not use the working backward problem solving strategy were in the high ...