maths problem solving cycle

Teaching Problem Solving in Math

• Freebies , Math , Planning

Every year my students can be fantastic at math…until they start to see math with words. For some reason, once math gets translated into reading, even my best readers start to panic. There is just something about word problems, or problem-solving, that causes children to think they don’t know how to complete them.

Every year in math, I start off by teaching my students problem-solving skills and strategies. Every year they moan and groan that they know them. Every year – paragraph one above. It was a vicious cycle. I needed something new.

I put together a problem-solving unit that would focus a bit more on strategies and steps in hopes that that would create problem-solving stars.

The Problem Solving Strategies

First, I wanted to make sure my students all learned the different strategies to solve problems, such as guess-and-check, using visuals (draw a picture, act it out, and modeling it), working backward, and organizational methods (tables, charts, and lists). In the past, I had used worksheet pages that would introduce one and provide the students with plenty of problems practicing that one strategy. I did like that because students could focus more on practicing the strategy itself, but I also wanted students to know when to use it, too, so I made sure they had both to practice.

I provided students with plenty of practice of the strategies, such as in this guess-and-check game.

There’s also this visuals strategy wheel practice.

I also provided them with paper dolls and a variety of clothing to create an organized list to determine just how many outfits their “friend” would have.

Then, as I said above, we practiced in a variety of ways to make sure we knew exactly when to use them. I really wanted to make sure they had this down!

Anyway, after I knew they had down the various strategies and when to use them, then we went into the actual problem-solving steps.

The Problem Solving Steps

I wanted students to understand that when they see a story problem, it isn’t scary. Really, it’s just the equation written out in words in a real-life situation. Then, I provided them with the “keys to success.”

S tep 1 – Understand the Problem.   To help students understand the problem, I provided them with sample problems, and together we did five important things:

• read the problem carefully
• restated the problem in our own words
• crossed out unimportant information
• circled any important information
• stated the goal or question to be solved

We did this over and over with example problems.

Once I felt the students had it down, we practiced it in a game of problem-solving relay. Students raced one another to see how quickly they could get down to the nitty-gritty of the word problems. We weren’t solving the problems – yet.

Then, we were on to Step 2 – Make a Plan . We talked about how this was where we were going to choose which strategy we were going to use. We also discussed how this was where we were going to figure out what operation to use. I taught the students Sheila Melton’s operation concept map.

We talked about how if you know the total and know if it is equal or not, that will determine what operation you are doing. So, we took an example problem, such as:

Sheldon wants to make a cupcake for each of his 28 classmates. He can make 7 cupcakes with one box of cupcake mix. How many boxes will he need to buy?

We started off by asking ourselves, “Do we know the total?” We know there are a total of 28 classmates. So, yes, we are separating. Then, we ask, “Is it equal?” Yes, he wants to make a cupcake for EACH of his classmates. So, we are dividing: 28 divided by 7 = 4. He will need to buy 4 boxes. (I actually went ahead and solved it here – which is the next step, too.)

Step 3 – Solving the problem . We talked about how solving the problem involves the following:

• taking our time
• working the problem out
• showing all our work
• estimating the answer
• using thinking strategies

We talked specifically about thinking strategies. Just like in reading, there are thinking strategies in math. I wanted students to be aware that sometimes when we are working on a problem, a particular strategy may not be working, and we may need to switch strategies. We also discussed that sometimes we may need to rethink the problem, to think of related content, or to even start over. We discussed these thinking strategies:

• switch strategies or try a different one
• rethink the problem
• think of related content
• decide if you need to make changes
• check your work
• but most important…don’t give up!

To make sure they were getting in practice utilizing these thinking strategies, I gave each group chart paper with a letter from a fellow “student” (not a real student), and they had to give advice on how to help them solve their problem using the thinking strategies above.

Finally, Step 4 – Check It.   This is the step that students often miss. I wanted to emphasize just how important it is! I went over it with them, discussing that when they check their problems, they should always look for these things:

• compare your answer to your estimate
• check for reasonableness
• check your calculations
• add the units
• restate the question in the answer
• explain how you solved the problem

Then, I gave students practice cards. I provided them with example cards of “students” who had completed their assignments already, and I wanted them to be the teacher. They needed to check the work and make sure it was completed correctly. If it wasn’t, then they needed to tell what they missed and correct it.

To demonstrate their understanding of the entire unit, we completed an adorable lap book (my first time ever putting together one or even creating one – I was surprised how well it turned out, actually). It was a great way to put everything we discussed in there.

Once we were all done, students were officially Problem Solving S.T.A.R.S. I just reminded students frequently of this acronym.

Stop – Don’t rush with any solution; just take your time and look everything over.

Think – Take your time to think about the problem and solution.

Act  – Act on a strategy and try it out.

Review – Look it over and see if you got all the parts.

Wow, you are a true trooper sticking it out in this lengthy post! To sum up the majority of what I have written here, I have some problem-solving bookmarks FREE to help you remember and to help your students!

You can grab these problem-solving bookmarks for FREE by clicking here .

You can do any of these ideas without having to purchase anything. However, if you are looking to save some time and energy, then they are all found in my Math Workshop Problem Solving Unit . The unit is for grade three, but it  may work for other grade levels. The practice problems are all for the early third-grade level.

• freebie , Math Workshop , Problem Solving

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• The Three Stages of the Problem-Solving Cycle

Essentially every problem-solving heuristic in mathematics goes back to George Polya’s How to Solve It ; my approach is no exception. However, this cyclic description might help to keep the process cognitively present.

A few months ago, I produced a video describing this the three stages of the problem-solving cycle: Understand, Strategize, and Implement. That is, we must first understand the problem, then we think of strategies that might help solve the problem, and finally we implement those strategies and see where they lead us. During two decades of observing myself and others in the teaching and learning process, I’ve noticed that the most neglected phase is often the first one—understanding the problem.

The Three Stages Explained

• What am I looking for?
• What is the unknown?
• Do I understand every word and concept in the problem?
• Am I familiar with the units in which measurements are given?
• Is there information that seems missing?
• Is there information that seems superfluous?
• Is the source of information bona fide? (Think about those instances when a friend gives you a puzzle to solve and you suspect there’s something wrong with the way the puzzle is posed.)
• Logical reasoning
• Pattern recognition
• Working backwards
• Adopting a different point of view
• Considering extreme cases
• Solving a simpler analogous problem
• Organizing data
• Making a visual representation
• Accounting for all possibilities
• Intelligent guessing and testing

I have produced videos explaining each one of these strategies individually using problems we have solved at the Chapel Hill Math Circle.

• Implementing : We now implement our strategy or set of strategies. As we progress, we check our reasoning and computations (if any). Many novice problem-solvers make the mistake of “doing something” before understanding (or at least thinking they understand) the problem. For instance, if you ask them “What are you looking for?”, they might not be able to answer. Certainly, it is possible to have an incorrect understanding of the problem, but that is different from not even realizing that we have to understand the problem before we attempt to solve it!

As we implement our strategies, we might not be able to solve the problem, but we might refine our understanding of the problem. As we refine our understanding of the problem, we can refine our strategy. As we refine our strategy and implement a new approach, we get closer to solving the problem, and so on. Of course, even after several iterations of this cycle spanning across hours, days, or even years, one may still not be able to solve a particular problem. That’s part of the enchanting beauty of mathematics.

I invite you to observe your own thinking—and that of your students—as you move along the problem-solving cycle!

[1] Problem-Solving Strategies in Mathematics , Posamentier and Krulik, 2015.

About the author: You may contact Hector Rosario at [email protected].

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Building fluency through problem solving

Editor’s Note:

This is an updated version of a blog post published on January 13, 2020.

Problem solving builds fluency and fluency builds problem solving. How can you help learners make the most of this virtuous cycle and achieve mastery?

Fluency. It’s so important that I have written not one , not two , but three blog posts on the subject. It’s also one of the three key aims for the national curriculum.

It’s a common dilemma. Learners need opportunities to apply their knowledge in solving problems and reasoning (the other two NC aims), but can’t reason or solve problems until they’ve achieved a certain level of fluency.

Instead of seeing this as a catch-22, think of fluency and problem solving as a virtuous cycle — working together to help learners achieve true mastery.

Supporting fluency when solving problems

Fluency helps children spot patterns, make conjectures, test them out, create generalisations, and make connections between different areas of their learning — the true skills of working mathematically. When learners can work mathematically, they’re better equipped to solve problems.

But what if learners are not totally fluent? Can they still solve problems? With the right support, problem solving helps learners develop their fluency, which makes them better at problem solving, which develops fluency…

Here are ways you can support your learners’ fluency journey.

Don’t worry about rapid recall

What does it mean to be fluent? Fluency means that learners are able to recall and use facts in a way that is accurate, efficient, reliable, flexible and fluid. But that doesn’t mean that good mathematicians need to have super-speedy recall of facts either.

Putting pressure on learners to recall facts in timed tests can negatively affect their ability to solve problems. Research shows that for about one-third of students, the onset of timed testing is the beginning of maths anxiety . Not only is maths anxiety upsetting for learners, it robs them of working memory and makes maths even harder.

Just because it takes a learner a little longer to recall or work out a fact, doesn’t mean the way they’re working isn’t becoming accurate, efficient, reliable, flexible and fluid. Fluent doesn’t always mean fast, and every time a learner gets to the answer (even if it takes a while), they embed the learning a little more.

Give learners time to think and reason

Psychologist Daniel Willingham describes memory as “the residue of thought”. If you want your learners to become fluent, you need to give them opportunities to think and reason. You can do this by looking for ways to extend problems so that learners have more to think about.

Here’s an example: what is 6 × 7 ? You could ask your learners for the answer and move on, but why stop there? If learners know that 6 × 7 = 42 , how many other related facts can they work out from this? Or if they don’t know 6 × 7 , ask them to work it out using facts they do know, like (5 × 7) + (1 × 7) , or (6 × 6) + (1 × 6) ?

Spending time exploring problems helps learners to build fluency in number sense, recognise patterns and see connections, and visualise — the three key components of problem solving.

Developing problem solving when building fluency

Learners with strong problem-solving skills can move flexibly between different representations, recognising and showing the links between them. They identify the merits of different strategies, and choose from a range of different approaches to find the one most appropriate for the maths problem at hand.

So, what type of problems should you give learners when they are still building their fluency? The best problem-solving questions exist in a Goldilocks Zone; the problems are hard enough to make learners think, but not so hard that they fail to learn anything.

Here’s how to give them opportunities to develop problem solving.

Centre problems around familiar topics

Learners can develop their problem-solving skills if they’re actively taught them and are given opportunities to put them into practice. When our aim is to develop problem-solving skills, it’s important that the mathematical content isn’t too challenging.

Asking learners to activate their problem-solving skills while applying new learning makes the level of difficulty too high. Keep problems centred around familiar topics (this can even be content taught as long ago as two years previously).

Not only does choosing familiar topics help learners practice their problem-solving skills, revisiting topics will also improve their fluency.

Keep the focus on problem solving, not calculation

What do you want learners to notice when solving a problem? If the focus is developing problem-solving skills, then the takeaway should be the method used to answer the question.

If the numbers involved in a problem are ‘nasty’, learners might spend their limited working memory on calculating and lose sight of the problem. Chances are they’ll have issues recalling the way they solved the problem. On top of that, they’ll learn nothing about problem-solving strategies.

It’s important to make sure that learners have a fluent recall of the facts needed to solve the problem. This way, they can focus on actually solving it rather than struggling to recall facts. To understand the underlying problem-solving strategies, learners need to have the processing capacity to spot patterns and make connections.

The ultimate goal of teaching mathematics is to create thinkers. Making the most of the fluency virtuous cycle helps learners to do so much more than just recall facts and memorise procedures. In time, your learners will be able to work fluently, make connections, solve problems, and become true mathematical thinkers.

Jo Boaler (2014). Research Suggests that Timed Tests Cause Math Anxiety. Teaching Children Mathematics , 20(8), p.469.

Willingham, D. (2009). Why don’t students like school?: A Cognitive Scientist Answers Questions About How the Mind Works and What It Means for Your Classroom. San Francisco: Jossey-Bass.

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Problem-solving cycle.

The Problem-Solving Cycle (PSC) is a National Science Foundation funded project that has developed a research-based professional development (PD) model.  This model is highly adaptable and can be specifically focused on problems of practice that are of interest to the participating teachers and administrators.  Additionally, it can be tailored to highlight federal, state, district, and school-based initiatives that are ever-changing and ongoing in the life of a teacher.

The PSC project is a research-practice partnership with the San Francisco Unified School District.  The current focus is on creating teacher leaders in middle school math classrooms and studying the effect on student learning.  

PSC Project Products:

Borko, H., Carlson, J., Jarry-Shore, M., Barnes, E., & Ellsworth, A. (2017, May). All students & teachers as math learners: A partnership to refine and implement two interconnected models. Presented at Stanford University’s CSET’s Pondering Excellence in Teaching Talk Series, Stanford, CA.

Borko, H., Carlson, J., Deutscher, R., & Ryan, J. (2018, May). A research-practice partnership to build district capacity. Video presented at 2018 STEM For All Video Showcase. http://stemforall2018.videohall.com/presentations/1299  

Borko, H. (2021 August). The Problem-Solving Cycle and Teacher Leadership Preparation Program: Developing and Researching a Model for Bringing Mathematics Professional Development to Scale . Research Seminar [Zoom] presented at IPN Leibniz Institute for Science and Mathematics Education, University of Kiel. 

Borko, H., Carlson, J., Mangram, C., Anderson, R., Fong, A., Million, S., Mozenter, S., & Villa, A. M. (2017). The role of video-based discussion in model for preparing professional development leaders. International Journal of STEM Education, 4 (1), 1-15.

Borko, H., Carlson, J., Deutscher, R., Boles, K. L., Delaney, V., Fong, A., Jarry-Shore, M., Malamut, J., Million, S., Mozenter, S., & Villa, A. M. (2021). Learning to Lead: an Approach to Mathematics Teacher Leader Development. International Journal of Science and Mathematics Education , 1-23.

Conference Presentations

Borko, H. (2015, February). Design-based implementation research in schools: Benefits & challenges . Paper presented at AACTE, Washington, D.C.

Borko, H., & Carlson, J. (2016, April) Design-based implementation research: adapting a professional development leadership model with a school district” Paper presented at AERA in a symposium entitled A Behind-the-Scenes Look at Effective Video-Based Professional Development , Washington, D.C.

Borko, H. (2016, June). Preparing mathematics teachers to facilitate the problem-solving cycle professional development . Paper presented at the Symposium and Workshop on Video Resources for Mathematics Teacher Development at the Weizmann Institute, Rehovot, Israel.

Mozenter, S. (2017, February). Video-based discussions: Meeting the multiple demands of PD for content teachers serving English language learners. Presented at National Association for Bilingual Education, Dallas, TX.

Borko, H., & Villa III, A. M. (2017, March). Facilitating Video-Based Mathematics Professional Development. Presented at Teacher Development Group Leadership Seminar, Portland, OR.

Villa III, A. M., & Jarry-Shore, M. (2017, March). Facilitating video-based mathematics professional development. Research symposium at National Council of Teachers of Mathematics Research Conference, San Antonio, TX.

Carlson, J., Jarry-Shore, M., Barnes, E., & Ellsworth, A. (2017, March). All students & teachers as math learners: A partnership to refine and implement two interconnected models.   Presented at Stanford-SFUSD Partnership Annual Meeting, Stanford, CA.

Jarry-Shore, M., Fong, A., Dyer, E., Gomez Zaccarelli, F., & Borko, H. (2018, February).  Video for equity: Designing video-based discussions of student authority.  Presentation at Association of Mathematics Teacher Education, Houston, TX.

Fong, A., Dyer, E., & Gomez Zaccarelli, F. (2018, February).  A shared vision for teacher improvement: Adapting professional development for local context by leveraging district-developed tools.  Presentation at Association of Mathematics Teacher Education, Houston, TX.

Mozenter, S., Gomez Zaccarelli, F., & Ellsworth, A. (2018, February ).  Video-based discussions in service of student agency, authority, and identity. Presentation at the Association of Teacher Education, Las Vegas, NV.

Mozenter, S., Ellsworth, A., & Gomez Zaccarelli, F. (2018, March). Video-based discussions in service of student agency, authority, & identity. Presentation at the American Association of Colleges for Teacher Education, Baltimore. MD.

Borko, H., & Villa III, A. M. (2018, March ). Building district capacity to address student access & equity: A research-practice partnership to develop teacher leaders. Presentation at the Teacher Development Group Leadership Seminar, Portland, OR.

Borko, H., Carlson, J., & Treviño, E. (2018, April).  A research-practice partnership to develop district capacity: Learning with & from each other.  Paper presented at the American Educational Research Association, New York, NY. 

Mozenter, S., Borko, H., & Jarry-Shore, M. (2018, June). Complicating the connection: Immigrant-background teachers . Paper presented at Teaching & Teacher Education Special Interest Group of the European Association for Research on Learning and Instruction, Kristiansaand, Norway.

Treviño, E. Brown, A., Villa III, A.M., & Borko, H. (2018, November).  Deconstructing student math content knowledge and groupwork through video-based discussion. Presentation at California Mathematics Council - Northern Section Conference Asilomar, Pacific Grove, CA.

Jarry-Shore, M. (2018, November ). The in-the-moment noticing of the novice mathematics teacher. Paper and presentation at the North American chapter of the International Group for the Psychology of Mathematics Education, Greenville, SC.

Villa III, A.M., & Boles, K. (2019, February).  Actualizing agency, authority, identity, and access to content in two contrasting cases of mathematical groupwork . Presented at Association of Mathematics Teacher Education, Orlando, FL.

Borko, H., & Villa III, A.M. (2019, February/March). Building teachers’ capacity to promote students’ access to rigorous and meaningful mathematics through video-based discussions. Presentation at the Teacher Development Group Leadership Seminar, Portland, OR.

Gomez Zaccarelli, F., Villa III, A.M., Mozenter, S., Boles, K., Deutscher, R., Borko, H., & Carlson, J. (2019, April).  How students are oriented toward a mathematical task and their peers: Access to content, agency, authority, and identity. Paper presented at the American Educational Research Association, Toronto, Canada. 

Mozenter, S., & Borko, H. (2019, April ). “ Not many people ask me this kind of question.” Three contrasting cases of immigrant-background teachers . Paper presented at the American Educational Research Association, Toronto, Canada.

Borko, H., Carlson, J., & Deutscher, R. (2019,  April ). Learning environments to support teacher leaders’ learning to lead video-based discussions. Poster presented in the structured poster session at the American Educational Research Association, Toronto, Canada .

Villa III, A.M., Boles, K.L., & Borko, H. (2019, November ).  Teacher leader learning through participation in and facilitation of professional development addressing problems of practice . Paper and presentation at the North American chapter of the International Group for the Psychology of Mathematics Education, St. Louis, MO.

Boles, K. L., Jarry-Shore, M., Muro Villa III, A., Malamut, J., & Borko, H. (2020, June). Building capacity via facilitator agency: Tensions in implementing an adaptive model of professional development. In M. Gresalfi, & I. S. Horn (Eds.),  The Interdisciplinarity of the Learning Sciences, 14th International Conference of the Learning Sciences (ICLS)  (pp. 2585-2588). Nashville, TN: International Society of the Learning Sciences. 

Jarry-Shore, M., & Allen, T. (2020, December). Noticing Struggle to Support Student Understanding [Conference Presentation]. California Mathematics Council - North Conference, Pacific Grove, CA, United States.

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PROBLEM SOLVING – Blake’s Guide to Maths Problem Solving

This guide is a vital tool for all middle and upper primary students who want to be successful maths problem solvers. The book is divided into three sections.

• Why Solve Problems? Explains which personal skills help you solve problems and what a problem solving cycle looks like.
• Common Problem Solving Strategies The 8 suggested strategies are: Visualise it, Make a table or graph, Guess and check, Break it into smaller parts, Work backwards, Look for a pattern, Eliminate possibilities.
• Types of Problems More than 140 sample problems include 1-step, 2-step, more than 2-step, multiple choice and open-ended problems.

The book includes detailed, suggested strategies, as well as TRY THIS and CHALLENGE activities to test your understanding. Answers are at the back of the book. It is suitable for students, teachers and parents working with Stage 2 or 3 students. This Guide is packed full of easy-to-understand explanations, real-life photographs and graphics. The book includes curriculum correlation charts. The book also showcases 7 outstanding male and female problem-solvers from across the world.

Click here to buy a copy directly from the publisher or to view sample pages.

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Overview of the Problem-Solving Mental Process

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Rachel Goldman, PhD FTOS, is a licensed psychologist, clinical assistant professor, speaker, wellness expert specializing in eating behaviors, stress management, and health behavior change.

• Identify the Problem
• Define the Problem
• Form a Strategy
• Organize Information
• Allocate Resources
• Monitor Progress
• Evaluate the Results

Frequently Asked Questions

Problem-solving is a mental process that involves discovering, analyzing, and solving problems. The ultimate goal of problem-solving is to overcome obstacles and find a solution that best resolves the issue.

The best strategy for solving a problem depends largely on the unique situation. In some cases, people are better off learning everything they can about the issue and then using factual knowledge to come up with a solution. In other instances, creativity and insight are the best options.

It is not necessary to follow problem-solving steps sequentially, It is common to skip steps or even go back through steps multiple times until the desired solution is reached.

In order to correctly solve a problem, it is often important to follow a series of steps. Researchers sometimes refer to this as the problem-solving cycle. While this cycle is portrayed sequentially, people rarely follow a rigid series of steps to find a solution.

The following steps include developing strategies and organizing knowledge.

1. Identifying the Problem

While it may seem like an obvious step, identifying the problem is not always as simple as it sounds. In some cases, people might mistakenly identify the wrong source of a problem, which will make attempts to solve it inefficient or even useless.

Some strategies that you might use to figure out the source of a problem include :

• Asking questions about the problem
• Breaking the problem down into smaller pieces
• Looking at the problem from different perspectives
• Conducting research to figure out what relationships exist between different variables

2. Defining the Problem

After the problem has been identified, it is important to fully define the problem so that it can be solved. You can define a problem by operationally defining each aspect of the problem and setting goals for what aspects of the problem you will address

At this point, you should focus on figuring out which aspects of the problems are facts and which are opinions. State the problem clearly and identify the scope of the solution.

3. Forming a Strategy

After the problem has been identified, it is time to start brainstorming potential solutions. This step usually involves generating as many ideas as possible without judging their quality. Once several possibilities have been generated, they can be evaluated and narrowed down.

The next step is to develop a strategy to solve the problem. The approach used will vary depending upon the situation and the individual's unique preferences. Common problem-solving strategies include heuristics and algorithms.

• Heuristics are mental shortcuts that are often based on solutions that have worked in the past. They can work well if the problem is similar to something you have encountered before and are often the best choice if you need a fast solution.
• Algorithms are step-by-step strategies that are guaranteed to produce a correct result. While this approach is great for accuracy, it can also consume time and resources.

Heuristics are often best used when time is of the essence, while algorithms are a better choice when a decision needs to be as accurate as possible.

4. Organizing Information

Before coming up with a solution, you need to first organize the available information. What do you know about the problem? What do you not know? The more information that is available the better prepared you will be to come up with an accurate solution.

When approaching a problem, it is important to make sure that you have all the data you need. Making a decision without adequate information can lead to biased or inaccurate results.

5. Allocating Resources

Of course, we don't always have unlimited money, time, and other resources to solve a problem. Before you begin to solve a problem, you need to determine how high priority it is.

If it is an important problem, it is probably worth allocating more resources to solving it. If, however, it is a fairly unimportant problem, then you do not want to spend too much of your available resources on coming up with a solution.

At this stage, it is important to consider all of the factors that might affect the problem at hand. This includes looking at the available resources, deadlines that need to be met, and any possible risks involved in each solution. After careful evaluation, a decision can be made about which solution to pursue.

6. Monitoring Progress

After selecting a problem-solving strategy, it is time to put the plan into action and see if it works. This step might involve trying out different solutions to see which one is the most effective.

It is also important to monitor the situation after implementing a solution to ensure that the problem has been solved and that no new problems have arisen as a result of the proposed solution.

Effective problem-solvers tend to monitor their progress as they work towards a solution. If they are not making good progress toward reaching their goal, they will reevaluate their approach or look for new strategies .

7. Evaluating the Results

After a solution has been reached, it is important to evaluate the results to determine if it is the best possible solution to the problem. This evaluation might be immediate, such as checking the results of a math problem to ensure the answer is correct, or it can be delayed, such as evaluating the success of a therapy program after several months of treatment.

Once a problem has been solved, it is important to take some time to reflect on the process that was used and evaluate the results. This will help you to improve your problem-solving skills and become more efficient at solving future problems.

A Word From Verywell​

It is important to remember that there are many different problem-solving processes with different steps, and this is just one example. Problem-solving in real-world situations requires a great deal of resourcefulness, flexibility, resilience, and continuous interaction with the environment.

Get Advice From The Verywell Mind Podcast

Hosted by therapist Amy Morin, LCSW, this episode of The Verywell Mind Podcast shares how you can stop dwelling in a negative mindset.

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You can become a better problem solving by:

• Practicing brainstorming and coming up with multiple potential solutions to problems
• Being open-minded and considering all possible options before making a decision
• Breaking down problems into smaller, more manageable pieces
• Asking for help when needed
• Researching different problem-solving techniques and trying out new ones
• Learning from mistakes and using them as opportunities to grow

It's important to communicate openly and honestly with your partner about what's going on. Try to see things from their perspective as well as your own. Work together to find a resolution that works for both of you. Be willing to compromise and accept that there may not be a perfect solution.

Take breaks if things are getting too heated, and come back to the problem when you feel calm and collected. Don't try to fix every problem on your own—consider asking a therapist or counselor for help and insight.

If you've tried everything and there doesn't seem to be a way to fix the problem, you may have to learn to accept it. This can be difficult, but try to focus on the positive aspects of your life and remember that every situation is temporary. Don't dwell on what's going wrong—instead, think about what's going right. Find support by talking to friends or family. Seek professional help if you're having trouble coping.

Davidson JE, Sternberg RJ, editors.  The Psychology of Problem Solving .  Cambridge University Press; 2003. doi:10.1017/CBO9780511615771

Sarathy V. Real world problem-solving .  Front Hum Neurosci . 2018;12:261. Published 2018 Jun 26. doi:10.3389/fnhum.2018.00261

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Supporting critical numeracy and maths skills in teaching and learning

In today’s article, Dave Tout, Justine Sakurai and Carly Sawatzki discuss numeracy and its relationship with mathematics, and the importance of real-world contexts. They’ll also share a problem-solving cycle to help students develop their skills and a classroom example of health numeracy, using trampolining as a focus for mathematical investigation .

Over the last couple of years, we have worked collaboratively to help shape and write Victorian senior secondary curricula that better supports the development of critical numeracy and maths skills in our school students. This is the first of two articles where we will share with you some of the research and theory that guided us, our thoughts about the purpose of education, and how these ideas influenced how we wrote those curriculum frameworks.

This first article will describe the background to our approach, and the second article will describe how we have attempted to espouse this in a curriculum document.

But first, what is, or are, our starting points? There are a few key elements behind our approach, including that:

• numeracy is the use and application of mathematical knowledge in context;
• numeracy is a social practice, which means there is always a clear purpose for learning that is connected to the real world;
• there is an implicit numeracy demand within most real-world problems; and,
• underpinning all this is a problem solving approach.

What is numeracy (and its relationships with mathematics)?

From our perspective, numeracy surrounds us in our everyday lives. Numeracy is about using mathematics to make sense of the world and applying mathematics in a context for a social purpose. For most young people and adults, numeracy gives meaning to mathematics, and mathematical knowledge and skills contribute to efficient and critical numeracy.

Students need a range of mathematical knowledge, skills, understandings and dispositions to solve problems in real contexts across personal, further learning, work, and community settings. To become numerate you need to know some mathematics. As Lynn Steen eloquently said:

...numeracy is not the same as mathematics, nor is it an alternative to mathematics. Today's students need both mathematics and numeracy. Whereas mathematics asks students to rise above context, quantitative literacy is anchored in real data that reflect engagement with life's diverse contexts and situations. (Steen, 2001, p.10)

Numeracy is not just about numbers and arithmetic. The mathematical knowledge and skills needed by all students includes number and quantity, measurement, shape, dimensions and directions, data and chance, and mathematical relationships and thinking. It also includes the ability to dip into your toolkit and choose and use the most appropriate analogue tools and digital technologies.

An Australian model of numeracy that we believe illustrates this perspective well is shown in Figure 1 below. This model incorporates four dimensions of contexts, mathematical knowledge, tools, and dispositions that are embedded in a critical orientation to using mathematics.

Why are real-world contexts so important?

Students often report that the mathematics they encounter at school feels disconnected from the real world. They express their frustration via the question, ‘When am I going to use this?’

Curriculum writers certainly intend that teachers bring their curriculum to life through contextualised lessons that connect with students’ real-world experiences. The complex challenges of modern life and work necessitate that schools deliver contextualised learning opportunities – students need higher levels of mathematics and numeracy than ever before and they need practise applying mathematics and numeracy to a range of familiar and unfamiliar authentic problems and issues (AAMT & AiGroup 2014; Binkley et al., 2012; FYA 2017; Gravemeijer et al., 2017).

As students become numerate, they develop the ability to make considered, mathematically-informed decisions, whether they be related to personal financial matters, planning travel arrangements, understanding and interpreting big data such as with the current COVID-19 epidemic, following instructions about a health or medical matter, or understanding the personal and social implications of problematic gambling.

Another of our key underpinning beliefs is that the exploration of real-world issues and problems is more valuable, satisfying and useful for students than the too frequent, often meaningless and repetitive practise of standard mathematical facts, procedures and processes. If students have little experience grappling with the messiness of real-world situations and problems, and if they can only apply mathematical procedures when problems are packaged in very familiar, structured ways (like in traditional maths classrooms and textbooks), then how can we expect them to value, see, use and apply maths in the world outside the classroom?

The AAMT and AiGroup research project referenced above documented this, with one of the teachers involved commenting on this disconnect:

This is one of the most interesting aspects/concepts of this project. The relationship between workplace mathematical skills and school mathematics could be described as ‘distant’ at best. Teacher observation (AAMT & AiGroup, 2014)

Being illiterate is considered an appalling state in modern Australia, yet evidence of significant numbers of students, especially young women, exiting the education system innumerate does not receive the same attention (ABS, 2013; OECD, 2017).

We believe therefore that we, as maths educators, need to support our students to be able to engage with and problem solve when maths is embedded in real-world situations and contexts, and this includes within our maths classrooms, as well as across the curriculum.

An underpinning problem-solving cycle

We believe that students need to develop the skills to problem solve, to investigate and solve a problem where the mathematics is embedded within a real-world context. The contexts should be the starting point, and students need well-planned and guided experiences with a structured problem-solving cycle , so that they know how to move from the real-world context to the mathematical world and apply their mathematical knowledge to find answers and solutions to the problem at hand.

An important aspect of numeracy is the ability to critically reflect on, evaluate and review your outcomes, and finally to be able to communicate and report on what you did and found.

In our curriculum endeavours, our suggested problem-solving cycle is modelled and adapted from the one used in the OECD's Programme for International Student Assessment (PISA) mathematical literacy assessment framework (OECD, 2019). This is consistent with the Figure 1 model. Our model has four distinct components, as represented in Figure 2 below.

The four stages in the problem-solving cycle are:

• Formulate : where you need to identify, select, and interpret the mathematical information embedded in a real-world context and decide and plan what mathematics you need to use and what questions you might ask.
• Act on and use mathematics : in this stage you need to do the maths ­­– perform the mathematical actions and processes so you can complete the task; this includes the use of a range of tools and technologies.
• Evaluate and reflect : here you are expected to check and reflect on both the mathematical processes you used and the reasonableness of your results and outcomes, especially in relation to the real-world context.
• Communicate and repor t: finally, there’s no point to the activity and investigations if you don’t document and report on your outcomes and any results. Here you need to use a combination of informal and formal mathematical representations.

Below is an example of an investigation taken from the context of health and fitness that illustrates the directions that teaching and learning can take when a numeracy problem-solving cycle is used in practice.

A classroom example – health numeracy

Health and safety related contexts provide interesting and useful opportunities to develop numeracy. Mathematical data and evidence can inform an understanding of risks, costs and benefits associated with such things as:

• Health and exercise
• Vaccination
• Prescription medications
• Alcohol and other drugs of addiction
• Medicare and health insurance

When an individual is informed, they are able to make better personal choices. This is not only good for the individual, their family and community, but can limit costs to the economy.

An example of a health and exercise context that might interest young people is trampolining. The backyard trampoline is a great tool in promoting health and fitness, and commercial trampolining centres are popular amongst young people.

The mock media report below includes statistical and numerical representations and language that underpin risk assessment and behaviour choices, and provides a context from which to initially introduce and study the problem. The ability to engage critically with and determine the trustworthiness of health reports presented by politicians, medical experts, journalists and social media influencers is essential.

Students might discuss:

• What is the probability (overall risk) of a spinal or head injury from trampolining?
• Do the benefits of trampolining outweigh the risks?
• What can be done to prevent accidents and injuries?
• What advice should be given to parents and children?

Once they identify the issues, they can mathematise the questions by unpacking the key concepts of probability and statistics using knowledge of fractions and percentages.

Student thinking may be extended by comparing safety figures between commercial centres, or health data from 20 years ago to today. Once students have considered the mathematics, reflecting on their findings should help them situate the problem and decide if the mathematics makes sense in the context.

A lesson may consider both aspects of the trampoline debate; cause of injuries, and building health.

A lesson sequence on trampoline health for younger students might consider asking students to keep a journal over a week or a month – how many times did they jump on their own or a friend’s trampoline? Students could consider how long each jumping session lasted for. They could break it down into ‘guesstimates’ of how much time was doing tricks , and how much time was straight jumps. They could conduct a survey and ask other students too. Their data could be displayed as a visual proportion on a line to aid in student development of estimation and proportional reasoning.

For an investigation, ask your students to estimate how many trampoline jumps would equal a two kilometre run ? How would they work this out? Trampolining provides a real-life context that is relevant and applicable to students’ daily lives, through which they can learn using the problem-solving cycle.

There are many paths that may be taken as you and your students mathematise this problem. They may want to approach this problem using distance as a perspective by converting jumps into lengths? Another method may be for them to time how long it takes to run as opposed to jump? There are no limits to the approaches that may be employed in applying a mathematical lens to the problem. Be as creative and as physically active as you like when carrying out your mathematical investigations.

Once you have completed the mathematical tasking, the results must be looked at and interpreted within the context of the trampolining context. Ask your students to evaluate and reflect on their thinking: Has the question been answered and do the answers make sense in relation to the question? Finish by asking them to write up the results – get them to make a poster or a video.

Whatever angle you choose to take when designing problem-based tasks for your students, remember to keep it real, relatable, and relevant!

Ashby, K., Pointer, S., Eager, D., & Day, L. (2015). Australian trampoline injury patterns and trends. Australian and New Zealand journal of public health , 39 (5), 491-494. https://doi.org/10.1111/1753-6...

Australian Association of Mathematics Teachers (AAMT) & Australian Industry Group (AiGroup). (2014). Tackling the School–Industry Mathematics Divide . Commonwealth of Australia. https://www.chiefscientist.gov... (PDF, 355KB)

Australian Bureau of Statistics. (2013). Programme for the International Assessment of Adult Competencies (Catalogue No. 4228.0). https://www.abs.gov.au/

Binkley, M., Erstad, O., Herman, J., Raizen, S., Ripley, M., Miller-Ricci, M., & Rumble, M. (2012). Defining twenty-first century skills. In P. Griffin, B. McGaw, & E. Care (Eds.), Assessment and Teaching of 21st Century Skills (pp. 17-66). Springer.

Carbonell, R. (2018, September 15). Double bounced: Why jumpy insurers are hopping out of the trampoline business. ABC News . https://www.abc.net.au/news/2018-09-15/trampoline-boom-loses-its-bounce-with-insurers/10249630

Foundation for Young Australians. (2017). The New Basics: Big data reveals the skills young people need for the New Work Order . FYA. https://www.fya.org.au/wp-content/uploads/2016/04/The-New-Basics_Update_Web.pdf (PDF, 1.5MB)

Geiger, V., Goos, M., & Forgasz, H. (2015). A rich interpretation of numeracy for the 21st century: A survey of the state of the field. ZDM Mathematics Education, 47 (4), 531-548. https://doi.org/10.1007/s11858-015-0708-1

Goos, M., Geiger, V., & Dole, S. (2014). Transforming professional practice in numeracy teaching. In Y. Li, E. Silver, & S. Li (Eds.), Transforming Mathematics Instruction (pp. 81-102). Springer. https://doi.org/10.1007/978-3-319-04993-9_6

Gravemeijer, K., Stephan, M., Julie, C., Lin, F. L., & Ohtani, M. (2017). What mathematics education may prepare students for the society of the future? International Journal of Science and Mathematics Education , 15 (1), 105-123. https://doi.org/10.1007/s10763-017-9814-6

Michie, F. (2015, September 11). Rise in serious trampoline injuries in children worries trauma specialists, prompts new research. ABC News . https://www.abc.net.au/news/2015-09-11/rise-in-trampoline-injuries-worries-trauma-specialists/6766536

OECD. (2017). Building Skills for All in Australia: Policy Insights from the Survey of Adult Skills. OECD Skills Studies, OECD Publishing. https://doi.org/10.1787/9789264281110-en .

OECD. (2019). PISA 2018 Assessment and Analytical Framework. PISA, OECD Publishing. https://doi.org/10.1787/b25efab8-en .

Sharwood, L. N., Adams, S., Blaszkow, T., & Eager, D. (2018). Increasing injuries as trampoline parks expand within Australia: a call for mandatory standards. Australian and New Zealand Journal of Public Health , 42 (2), 153-156. https://doi.org/10.1111/1753-6405.12783

Steen, L. (2001). Mathematics and Numeracy: Two Literacies, One Language. The Mathematics Educator , (6)1, 10-16.

With a colleague, consider a problem-based task you’ve used in one of your own lessons. Was this problem real, relatable and relevant to your students, their context and experiences? Now, work together to design a new task for an upcoming lesson or topic area that meets these requirements.

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• Key Documents
• News/Events
• CPD Workshops 2017/2018
• CPD Workshops 2018/2019
• CPD Workshops 2019/2020
• CPD Workshops 2020/2021
• CPD Workshops 2021/2022
• CPD Workshop Day 2
• CPD Workshop Day 3
• CPD Workshops 2022/2023
• Elective Workshops
• STE(A)M in Junior Cycle
• Departmental Planning
• Learning Outcomes in Action
• Planning Resources
• room Resources

This section contains some initial information and resources to support you in engaging with the learning outcomes of the Specification for Junior Cycle Mathematics. Click on each title bar to view resources.

The unifying strand and making connections arrow_drop_down.

The video found here is from Youcubed at Stanford. Their main goal is to inspire, educate and empower teachers of mathematics, transforming the latest research on math learning into accessible and practical forms.

The components of mathematical proficiency - interactive poster arrow_drop_down.

Question Posing and Problem Formulation arrow_drop_down

Supporting students in developing their ability to pose questions and formulate problems in junior cycle mathematics is a self-guided interactive piece of cpd. this cpd has been designed to give junior cycle mathematics teachers the opportunity to consider how they can enable students to develop their question posing and problem formulation abilities. the anticipated time to complete this cpd is 45 minutes., two versions of the booklet have been made available for this option. for a hard copy of the booklet it is advisable to print the pdf version of the booklet. for a digital copy of the booklet it is advisable to download the word version of the booklet..

This resource has three sections. The first section considers the benefits of integrating question posing and problem formulation into pedagogical practice. The second section looks for teachers to engage with classroom strategies for developing students' ability to pose mathematical problems and questions. The third section considers ways to support students when defining problem statements as part of Mathematical Investigation (CBA 1).

To access each section from this page use the 'click to view' button. at the end of each section there is a hyperlink which opens the next section from within the form as an alternative to returning to this page..

L2LPs and Inclusive Education in Junior Cycle Mathematics - Interactive Resource arrow_drop_down

Welcome to this self-directed cpd on level 2 learning programmes (l2lps) and inclusive education for junior cycle mathematics. this resource has four sections. the first section provides an opportunity for you to consider some inclusive practices you have used to make the curriculum more accessible to students with special educational needs and to learn about the structure and features of l2lps. section two looks at some of the commonly asked questions that arise in relation to l2lps through an interactive quiz. section three provides an opportunity to examine a unit of learning designed for an inclusive approach to the incorporation of learning outcomes from both level 2 plus and the level 3 mathematics specification, and to examine a statistics task aligned to the key learning, ongoing assessment and evidence of learning outlined in the unit. in section four the principles of universal design for learning (udl) are introduced and further resources on udl are detailed., the approximate time for completing all four sections in this self-directed cpd is 45 minutes. no personal data or answers are being recorded or collected by junior cycle for teachers. before you begin please open or download the booklet which accompanies this self-directed cpd..

Unit of Learning Resources

The resources found here are the resources featured in section three of the self-directed cpd l2lps and inclusive education in junior cycle mathematics. these resources offer an opportunity to examine a unit of learning which incorporates learning outcomes for both level 2 and level 3 programmes and to examine a statistics task aligned to the key learning, ongoing assessment and evidence of learning outlined in the unit..

The task feartured here has been adapted with the kind permission of the JCT L1LPs/L2LPs Team.

Further resources, the resources found here are the resources featured in section four of the self-directed cpd 'l2lps and inclusive education in junior cycle mathematics'. the l2lps plu assessment is a downloadable spreadsheet. further resources for l2lps can be found here ..

Planning Interactive Resource arrow_drop_down

This self-guided interactive cpd is entitled planning for teaching, learning and assessment. it will give you the opportunity to consider how your school's plan can enable students to experience mathematics as a connected body of ideas. it can be used individually or as a subject department. the approaches, ideas and concepts presented are not intended to be definitive, and teachers could well form a different set of ideas or try alternative approaches. they may be useful to think about when considering your school's approach to planning. the anticipated time to complete the core part of this interactive cpd is 1 hour and 30 minutes., planning interactive booklet, math connections video, this resource is a downloadable word document which you will need to engage with this self-guided cpd planning for teaching, learning and assessment . use the 'click to view or download' button to access your own copy of this resource. the core parts of this document are: exploring concepts and ideas in junior cycle mathematics planning with concepts and ideas in junior cycle mathematics planning - next steps for our department.

The video found here is from Youcubed at Stanford . Their main goal is to inspire, educate and empower teachers of mathematics, transforming the latest research on math learning into accessible and practical forms. This video is featured as part of the Concepts and Connections portion of the interactive booklet of this self-guided CPD.

Year plan template with prompts, units of learning with prompts, the template found here can be used to map out an overview of first-year maths in your school there are prompt questions included which can be used to support consideration and discussion of plans as they are developed. these prompt questions may also support the ongoing review of plans..

The resource found here may be useful when creating or reviewing Units of Learning. There are prompt questions included which can be used to support consideration and discussionof units of learning as they are developed. These prompt questions may also support the ongoing review of units of learning.

Dynamic Mathematics Software - Desmos arrow_drop_down

Desmos is dynamic mathematics software for all levels of education that brings together a free suite of maths software tools, including a graphing calculator and geometry tool. the techology powers activities which can open up a world of possiblilites for students to explore concepts more deeply, collaborate with their peers on problem-solving, and apply knowlege creatively as mathematicians. you can access the free online maths software  here . the videos found in this section are to offer some guidance and support for teachers who may wish to use desmos in the junior cycle mathematics classroom., in this section you will find a sample of resources which are available on the desmos website. use the links embedded in each document to access these suggested resources..

Webinars Featuring Desmos

A recording of the jct mathematics team webinar 'delving into desmos with dr. dan meyer'  along with the webinar 'desmos to for junior cycle mathematics: using desmos to deepen student learning' can be found in this section. they may offer further guidance and support for teachers. , dynamic mathematics software - geogebra arrow_drop_down, geogebra is dynamic mathematics software for all levels of education that brings together geometry, algebra, spreadsheets, graphing, statistics and calculus in one easy-to-use package. geogebra's community includes a global community of teachers and students, enabling localized support for learning and improvements in mathematics education. you can access the free online math tools for graphing, geometry 3d and more here . the videos found in this section are to offer some guidance and support for teachers who may wish to use geogebra in the junior cycle mathematics classroom., in this section you will find a sample of resources which are available on the geogebra website. use the links embedded in each document to access these suggested resources..

Data Manipulation - CODAP arrow_drop_down

Codap is free open-source educational software for data analysis. it can be used by teachers and students in the classroom and at home. codap can be used with data that students have gathered themselves or with data from online sources such as cso  and censusatschool . codap also has pre-loaded data sets that you might like to explore. you can access the codap software and data sets   here . the videos found in this section are to offer some guidance and support for teachers who may wish to use the codap software in the junior cycle mathematics classroom., codap as a tool for teaching and learning mathematics, one teacher's approach, the purpose of this webinar is to offer an introduction to codap as a tool for teaching and learning mathematics. the webinar focused on two areas of support: 1. exploring the use of datasets on codap, both sourced and collected 2. exploring some of the functions available on codap, this next video was created for use with our online cluster cpd in mathematics. in the video john maddock, a teacher in etss wicklow, talks about how he uses codap in his classroom practice., digital learning resources arrow_drop_down, digital tools & apps for formative assessment in mathematics, digital manipulatives for teaching and learning in mathematics, formative assessment occurs during learning and offers the teacher and student information about how learning is progressing. it involves using assessment approaches by the teacher to better understand student learning and inform pedagogy during a lesson, and/or between lessons. this poster details some digital tools and apps which might be useful for formative assessment..

Digital manipulatives are useful learning and teaching tools that allow students to visualise and explore mathematical relationships, properties and facts across all strands of the mathematics specification. They can be used to support, challenge and extend student learning, through the use of open-ended tasks and investigations. This poster details some digital manipulatives which may be useful in a mathematics classroom.

Microsoft Forms for the Maths Classroom

The resources found here will demonstrate some practical ideas and uses for microsoft forms in the maths classroom. forms can be a useful tool, providing instant feedback on student learning. in this section we have highlighted auto correction, formative assessment, feedback and branching. similar features and functionality are available when using google forms..

Keyboard Shortcuts for Inputting Equations into Microsoft Office Applications

The video found here will demonstrate how to use keyboard shortcuts to input equations in microsoft office applications. for the purpose of this demonstration we have used microsoft word, however these keyboard shortcuts will work in other applications., onenote for the maths classroom, linking a google form to a google sheet, this resource is a downloadabe onenote notebook from our symposium held in 2019. the purpose of this onenote notebook is to demonstrate some practical ideas and uses for onenote in the maths classroom. for more supports in using onenote please visit the microsoft educator centre here ..

The video found here will demonstrate how to link a google form and a google sheet in order to collect and analyse data. To see how google forms and google sheets could be used for carrying out a semi-structured statistical investigation, review the presentation slides and page 17 of the booklet from the Mathematics CPD Workshops 2019/2020 available here .

How to measure angles using a protractor in a presentation, how to add geometry tools to a presentation, this resource is a downloadable presentation which is compatible with both microsoft powerpoint and google slides. to get full use of this resource you will need to download the presentation. use the 'click to download' button to do so. the purpose of the presentation is to demonstrate how to measure angles using a protractor in a presentation. there are a number of other geometry tools included at the end of the presentation which could be adapted for use in the classroom..

This resource is a step by step guide which demonstrates how to add a protractor, or any other geometry tool, to a PowerPoint presentation. To see how this could be used, refer to the presentation titled 'How to Measure Angles using a Protractor in a Presentation' which is also available in this section of the website.

Learning Tools - Immersive Reader

How to use an android device as a visualiser, immersive reader is a free tool that implements techniques to improve reading and writing regardless of age or ability. immersive reader can improve reading comprehension, increase fluency, help build confidence for emerging readers and offer text decoding solutions..

This resource is a step by step guide which demonstrates how to turn any Android device into a mobile visualiser for use in the maths classroom. A mobile visualiser can be used to promote class discussion by sharing/projecting student work. You will need an Android device, a Windows 10 PC and access to a wifi network to use this guide.

Interactive Applications for Formative Assessment

The resources found here will demonstrate some interactive applications which can be useful for providing students with formative feedback and formative assessment. these resources have been adapted with the kind permission of the jct mfl team..

Adobe Spark for the Maths Classroom

How to upload a video to youtube, adobe spark is an application which can be used to create social graphics and short videos. it could be useful as a way for students to communicate and demonstrate their learning in the maths classroom. this resource will demonstrate some of the functionality of adobe spark. this resource has been adapted with the kind permission of the jct mfl team..

This video demonstrates how to upload a video to YouTube. Hosted here with the kind permission of the JCT MFL team.

Engaging with online tutorials and courses for digital learning, online tutorials and courses for digital learning are widely available. we are highlighting two particularly useful sources for these tutorials and courses here. 1. the microsoft learn educator cente r is one of the world's largest educator social networks. 2. the google for education teacher center is a site for professional development, built by educators for educators. the documents in this section give an overview of what to expect, as a teacher, from these online digital learning supports..

Webinar Series arrow_drop_down

Desmos for junior cycle mathematics: using desmos to deepen student learning, the purpose of this webinar is to offer an introduction to desmos as a tool for teaching and learning mathematics. desmos is dynamic software for all levels of education that brings together a free suite of maths software tools, including a graphing calculator and geometry tool. further support materials for desmos are available in the resources section of our website., question posing for classroom-based assessment 1: mathematical investigation, in this webinar we are joined by a panel of teachers who share their experiences of, and advice for, supporting students with classroom-based assessment. the webinar begins with a taster of some of the strategies for question posing and problem formulation from the interactive cpd found here . some of the panel discussion includes: planning as a department for question posing, supporting students with posing questions and, developing and refining questions for investigation with students., in this section you will find excerpts from the webinar 'question posing for classroom-based assessment 1: mathematical investigation'., in this short video you will hear teachers discussing some of their ideas for departmental planning. they speak about how they plan to integrate question posing into their subject department plans incrementally for junior cycle mathematics., in this short video teachers discuss a variety of different ways that they support their students when refining their mathematical question(s)., in this short video you will hear teachers discuss how they support their students to develop posed questions. they speak about the importance of supporting students to make questions viable for mathematical investigation., in this short video teachers will share some advice they have for other teachers preparing for classroom-based assessment with their students., in this short video you will hear teachers discussing their experiences of classroom-based assessment in mathematics., delving into desmos with dr. dan meyer, based in california, dr. dan meyer is the chief academic officer with desmos. desmos is a free suite of maths software for students, available at desmos.com . during this workshop, dan discusses the interaction between pedagogy and digital learning technologies in mathematics., the purpose of this webinar is to offer an introduction to codap as a tool for teaching and learning mathematics. codap is free open-source education software for data analysis. the webinar focused on two areas of support:         1. exploring the use of datasets on codap, both sourced and collected                 2. exploring some of the functions available on codap further support materials for codap are available on our website in the data manipulation - codap section of this webpage., preparing for classroom-based assessment 2: statistical investigation & subject learning and assessment review meeting (slar), the purpose of this webinar is to offer some support for mathematics teachers preparing for classroom-based assessment 2: statistical investigation and the subject learning and assessment review meeting. the webinar is delivered in three parts:           1. teaching, learning and language of statistics           2. classroom-based assessment 2 and formative feedback           3. procedures for teachers before, during and after cba2 support documents that were mentioned in the webinar are available on our website under key documents , assessment and resources ., task selection and problem-solving in mathematics, in this webinar dr paddy johnson (university of limerick) discusses selection criteria for mathematical tasks and activities which develop students' ability to problem-solve., in this section you will find excerpts from the webinar 'task selection and problem-solving in mathematics'. each of the video excerpts below give an insight into areas for consideration when designing and/or selecting tasks for students., mathematics and being creative, the focus of this webinar is an interactive discussion around creativity in mathematics, mathematics classrooms and in teaching. there were three discussion points for the webinar:           1. what could creativity look like in the classroom           2. sharing and discussing perspectives on creativity.           3. fostering creativity in the classroom. we are grateful to all of the teachers that contributed to the discussion on the night with their own thought-provoking comments, questions and discussion points which were submitted via text and to all of the contributors that were willing to add to the discussion live on air., planning for teaching, learning and assessment: one school's approach, the resources in this section were developed for our webinar 'planning for teaching, learning and assessment: one school's approach. we would recommend that you view the webinar, available on this page, to see how the resources were developed and their potential for adaptation for your own school context. our thanks to the staff of etss wicklow for their contributions to these supports for teachers..

Preparing for Classroom-Based Assessment 1:​ Mathematical Investigation & Subject Learning and Assessment Review Meeting (SLAR)

This purpose of this webinar is to offer some support for mathematics teachers preparing for classroom-based assessment 1: mathematical investigation and the subject learning and assessment review meeting. the webinar is delivered in four parts:           1. setting the context for cba 1           2. supporting student learning before cba 1           3. supporting student learning during cba 1           4. preparing for, and participating in a slar meeting support documents that were mentioned in the webinar are available on our website under key documents , assessment and resources ..

Supporting Deeper Learning in the Mathematics Classroom

The format of this webinar was a live panel discussion which was focused on supporting deeper learning in the maths classroom. the discussion was prompted by research on the ideas of surface learning and deep learning in mathematics education, as well as questions and comments submitted by listeners live on the night. we were delighted to be joined by aidan fitzsimons, phd candidate in mathematics education in dcu, audrey byrne, mathematics teacher from lucan community college, dublin and peter fitton, mathematics teacher from bishopstown community school, cork. the main talking points were:           • surface learning and deep learning           • how do you create opportunity for deep learning           • deep learning and the junior cycle mathematics specification.

Planning for Teaching and Learning using the Mathematics Specification

The purpose of this webinar is to offer teachers some support for teachers in planning using the junior cycle mathematics specification. the webinar focused on two areas of support:           1. making connections within and between strands           2. creating units of learning and considering possible approaches to capturing our planning support documents that were mentioned in the webinar are available on our website under key documents , planning , assessment and resources ., engaging with the mathematics specification, this webinar is an introduction to the junior cycle mathematics specification. the talking points for the webinar were as follows:           1. what is the structure of the specification           2. working with and unpacking learning outcomes           3. beginning to plan for learning and teaching using our specification           4. assessment to support student learning support documents that were mentioned in the webinar are available on our website under key documents , wellbeing and resources ..

Algebra Through the Lens of Functions

Algebra through the lens of functions has been developed by the maths development team and is hosted here with their kind permission..

Podcasts arrow_drop_down

Tangents - a jct maths team podcast, tangents is a jct mathematics team podcast series available on the  junior cycle talks  podcast channel.  in this first series we spoke with a number of different people working in maths education to find out about creativity in the context of a maths classroom. to hear more podcasts from the  junior cycle talks  channel click here or search for us on soundcloud, spotify, apple podcasts or anywhere you listen to your podcasts.  .

Planning Resources arrow_drop_down

Planning for teaching, learning and assessment: one school's aproach.

Suggested Templates for Units of Learning

Further Planning Resources

SLAR Resources arrow_drop_down

Tasks & Learning Experiences arrow_drop_down

The task resource booklet, available to print or download here, contains a selection of mathematical tasks and activities which have been engaged with by teachers for the purposes of professional development. the tasks are suitable for use with students of junior cycle mathematics. teachers should use their professional judgement when considering the suitability of a mathematical task or activity and should adapt the task if necessary, to meet the needs and/or context of their students. tasks have been designed to be accessible, challenging and extendable, and adaptations should seek to maintain these design principles. a number of the tasks and activities from the booklet are also available as individual documents for your convenience in this section..

Resources for Assessment arrow_drop_down

Classroom based assessment.

Other Assessment Resources

Posters arrow_drop_down

Casio fx-83GTX Video Tutorials arrow_drop_down

Casio have a released a new version of their popular student calculator.  this new version has an updated setup menu and new and redesigned features. these videos are intended to support teachers in using this calculator in the classroom.  the setup menu is explained and the commonly used features: table and statistics are described through sample questions.  the new ratio feature is presented along with questions where this feature could be used by students. a short overview of using the memory is also provided. the casio fx-83gtx is on the state examinations commission’s list of approved calculators..

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Problem solving is an important skill which students should be encouraged to develop over a period of time. With careful planning it is possible to move from a classroom environment where individual lessons are devoted to problem-solving to one where the majority of the teaching and learning is done through problem solving. In this way students experience problem-solving on a daily basis, get into the habit of thinking about their maths and to applying knowledge they already have to tackle something new. This also helps them to see how different topics in maths are connected.

The documents below focus on teaching mathematical topics through problem-solving contexts and enquiry-oriented environments which are characterised by the teacher ‘helping’ students construct a deep understanding of mathematical ideas and processes by engaging them in doing mathematics: creating, conjecturing, exploring, testing, and verifying.

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Find the frequency distribution for ordinary English, and use it to help you crack the code.

Speed-time Problems at the Olympics

Have you ever wondered what it would be like to race against Usain Bolt?

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In which Olympic event does a human travel fastest? Decide which events to include in your Alternative Record Book.

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Rich mathematical tasks to develop powerful mathematical thinking

Flick through the eBooks for My Problem-Solving Journal and try out the sample problems

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Check out this short overview video of My Problem-Solving Journal! 

My Problem-Solving Journal for rich mathematical tasks

• Inspire a love of maths with real-life scenarios and problems that are relevant to children
• Deepen conceptual understanding and explore the big ideas of mathematics
• Challenge all children at their level with low threshold high ceiling tasks
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Explore one problem in depth each week

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Join teacher and author, Patrick Neary, to explore mathematical modeling in the primary classroom. See how seamlessly mathematical modeling can be emphasised through context-rich, open-ended problems, such as those found in My Problem-Solving Journal.

Join teacher and author, Elaine Dillion, to explore what problem-solving looks like for younger children . Examine how rich, meaningful tasks, such as those found in My Problem-Solving Journal, can be used to enrich children’s mathematical learning.

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My Problem-Solving Journal in Three Simple Steps

"It provides everything teachers need to help children in their class become critical thinkers and problem solvers"

Patrick Neary, experienced teacher and one of the authors of My Problem Solving Journal explains how MPSJ provides rich mathematical tasks to develop powerful mathematical thinking, and inspires a love of maths with real-life scenarios and problems that are relevant to children.

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All of our authors are currently teaching different classes.

Patrick Neary is the author of 5th & 6th Class, and Series Editor of the programme. He holds a M.Ed., specialising in Mathematics Education, from DCU, where he has also lectured part-time. Patrick has worked closely with Maths4All to promote best practice in Maths Education.

Grace Lynch is the author of 3rd & 4th Class. She holds a M.Ed., specialising in Mathematics Education, from DCU, and has supervised undergraduate dissertations for students specialising in Mathematics Education in Marino Institute.

Elaine Dillion is the author of 1st & 2nd Class. She holds a M.Ed. from Maynooth University, where her thesis focused on exploring an effective use of play in the teaching and learning of mathematics. Elaine is an active participant and contributor to Maths4All.

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The math problem that took nearly a century to solve: Secret to Ramsey numbers

Mathematicians unlock the secret to ramsey numbers.

We've all been there: staring at a math test with a problem that seems impossible to solve. What if finding the solution to a problem took almost a century? For mathematicians who dabble in Ramsey theory, this is very much the case. In fact, little progress had been made in solving Ramsey problems since the 1930s.

Now, University of California San Diego researchers Jacques Verstraete and Sam Mattheus have found the answer to r(4,t), a longstanding Ramsey problem that has perplexed the math world for decades.

What was Ramsey's problem, anyway?

In mathematical parlance, a graph is a series of points and the lines in between those points. Ramsey theory suggests that if the graph is large enough, you're guaranteed to find some kind of order within it -- either a set of points with no lines between them or a set of points with all possible lines between them (these sets are called "cliques"). This is written as r(s,t) where s are the points with lines and t are the points without lines.

To those of us who don't deal in graph theory, the most well-known Ramsey problem, r(3,3), is sometimes called "the theorem on friends and strangers" and is explained by way of a party: in a group of six people, you will find at least three people who all know each other or three people who all don't know each other. The answer to r(3,3) is six.

"It's a fact of nature, an absolute truth," Verstraete states. "It doesn't matter what the situation is or which six people you pick -- you will find three people who all know each other or three people who all don't know each other. You may be able to find more, but you are guaranteed that there will be at least three in one clique or the other."

What happened after mathematicians found that r(3,3) = 6? Naturally, they wanted to know r(4,4), r(5,5), and r(4,t) where the number of points that are not connected is variable. The solution to r(4,4) is 18 and is proved using a theorem created by Paul Erdös and George Szekeres in the 1930s.

Currently r(5,5) is still unknown.

A good problem fights back

Why is something so simple to state so hard to solve? It turns out to be more complicated than it appears. Let's say you knew the solution to r(5,5) was somewhere between 40-50. If you started with 45 points, there would be more than 10 234 graphs to consider!

"Because these numbers are so notoriously difficult to find, mathematicians look for estimations," Verstraete explained. "This is what Sam and I have achieved in our recent work. How do we find not the exact answer, but the best estimates for what these Ramsey numbers might be?"

Math students learn about Ramsey problems early on, so r(4,t) has been on Verstraete's radar for most of his professional career. In fact, he first saw the problem in print in Erdös on Graphs: His Legacy of Unsolved Problems, written by two UC San Diego professors, Fan Chung and the late Ron Graham. The problem is a conjecture from Erdös, who offered $250 to the first person who could solve it.

"Many people have thought about r(4,t) -- it's been an open problem for over 90 years," Verstraete said. "But it wasn't something that was at the forefront of my research. Everybody knows it's hard and everyone's tried to figure it out, so unless you have a new idea, you're not likely to get anywhere."

Then about four years ago, Verstraete was working on a different Ramsey problem with a mathematician at the University of Illinois-Chicago, Dhruv Mubayi. Together they discovered that pseudorandom graphs could advance the current knowledge on these old problems.

In 1937, Erdös discovered that using random graphs could give good lower bounds on Ramsey problems. What Verstraete and Mubayi discovered was that sampling from pseudo random graphs frequently gives better bounds on Ramsey numbers than random graphs. These bounds -- upper and lower limits on the possible answer -- tightened the range of estimations they could make. In other words, they were getting closer to the truth.

In 2019, to the delight of the math world, Verstraete and Mubayi used pseudorandom graphs to solve r(3,t). However, Verstraete struggled to build a pseudorandom graph that could help solve r(4,t).

He began pulling in different areas of math outside of combinatorics, including finite geometry, algebra and probability. Eventually he joined forces with Mattheus, a postdoctoral scholar in his group whose background was in finite geometry.

"It turned out that the pseudorandom graph we needed could be found in finite geometry," Verstraete stated. "Sam was the perfect person to come along and help build what we needed."

Once they had the pseudorandom graph in place, they still had to puzzle out several pieces of math. It took almost a year, but eventually they realized they had a solution: r(4,t) is close to a cubic function of t . If you want a party where there will always be four people who all know each other or t people who all don't know each other, you will need roughly t 3 people present. There is a small asterisk (actually an o) because, remember, this is an estimate, not an exact answer. But t 3 is very close to the exact answer.

The findings are currently under review with the Annals of Mathematics .

"It really did take us years to solve," Verstraete stated. "And there were many times where we were stuck and wondered if we'd be able to solve it at all. But one should never give up, no matter how long it takes."

Verstraete emphasizes the importance of perseverance -- something he reminds his students of often. "If you find that the problem is hard and you're stuck, that means it's a good problem. Fan Chung said a good problem fights back. You can't expect it just to reveal itself."

Verstraete knows such dogged determination is well-rewarded: "I got a call from Fan saying she owes me $250."

• Mathematics
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• Mathematical Modeling
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• Knot theory
• Massively multiplayer online game
• Supercomputer
• Algebraic geometry
• John von Neumann
• Scientific method

Story Source:

Materials provided by University of California - San Diego . Original written by Michelle Franklin. Note: Content may be edited for style and length.

Journal Reference :

• Sam Mattheus, Jacques Verstraete. The asymptotics of r(4,t) . Annals of Mathematics , 2024; 199 (2) DOI: 10.4007/annals.2024.199.2.8

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