critical thinking and problem solving skills used in numeracy program

Engaging Maths

Dr catherine attard, promoting creative and critical thinking in mathematics and numeracy.

• by cattard2017
• Posted on June 25, 2017

What is critical and creative thinking, and why is it so important in mathematics and numeracy education?

Numeracy is often defined as the ability to apply mathematics in the context of day to day life. However, the term ‘critical numeracy’ implies much more. One of the most basic reasons for learning mathematics is to be able to apply mathematical skills and knowledge to solve both simple and complex problems, and, more than just allowing us to navigate our lives through a mathematical lens, being numerate allows us to make our world a better place.

The mathematics curriculum in Australia provides teachers with the perfect opportunity to teach mathematics through critical and creative thinking. In fact, it’s mandated. Consider the core processes of the curriculum. The Australian Curriculum (ACARA, 2017), requires teachers to address four proficiencies : Problem Solving, Reasoning, Fluency, and Understanding. Problem solving and reasoning require critical and creative thinking (). This requirement is emphasised more heavily in New South wales, through the graphical representation of the mathematics syllabus content , which strategically places Working Mathematically (the proficiencies in NSW) and problem solving, at its core. Alongside the mathematics curriculum, we also have the General Capabilities , one of which is Critical and Creative Thinking – there’s no excuse!

Critical and creative thinking need to be embedded in every mathematics lesson . Why? When we embed critical and creative thinking, we transform learning from disjointed, memorisation of facts, to sense-making mathematics. Learning becomes more meaningful and purposeful for students.

How and when do we embed critical and creative thinking?

There are many tools and many methods of promoting thinking. Using a range of problem solving activities is a good place to start, but you might want to also use some shorter activities and some extended activities. Open-ended tasks are easy to implement, allow all learners the opportunity to achieve success, and allow for critical thinking and creativity. Tools such as Bloom’s Taxonomy and Thinkers Keys  are also very worthwhile tasks. For good mathematical problems go to the nrich website . For more extended mathematical investigations and a wonderful array of rich tasks, my favourite resource is Maths300   (this is subscription based, but well worth the money). All of the above activities can be used in class and/or for homework, as lesson starters or within the body of a lesson.

Will critical and creative thinking take time away from teaching basic concepts?

No, we need to teach mathematics in a way that has meaning and relevance, rather than through isolated topics. Therefore, teaching through problem-solving rather than for problem-solving. A classroom that promotes and critical and creative thinking provides opportunities for:

• higher-level thinking within authentic and meaningful contexts;
• complex problem solving;
• open-ended responses; and
• substantive dialogue and interaction.

Who should be engaging in critical and creative thinking?

Is it just for students? No! There are lots of reasons that teachers should be engaged with critical and creative thinking. First, it’s important that we model this type of thinking for our students. Often students see mathematics as black or white, right or wrong. They need to learn to question, to be critical, and to be creative. They need to feel they have permission to engage in exploration and investigation. They need to move from consumers to producers of mathematics.

Secondly, teachers need to think critically and creatively about their practice as teachers of mathematics. We need to be reflective practitioners who constantly evaluate our work, questioning curriculum and practice, including assessment, student grouping, the use of technology, and our beliefs of how children best learn mathematics.

Critical and creative thinking is something we cannot ignore if we want our students to be prepared for a workforce and world that is constantly changing. Not only does it equip then for the future, it promotes higher levels of student engagement, and makes mathematics more relevant and meaningful.

How will you and your students engage in critical and creative thinking?

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Critical thinking and problem solving skills used in numeracy program

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Critical thinking and problem solving skills used in

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Problem Solving and Critical Thinking

It means that, as far as a business is concerned, achieving an organisational objective Writers of symbolic accounts don’t just throw in minor details to confuse their readers – every detail must mean something. If you write two, three or more days in a row, you get even more points. Even the best readers need 8 or 10 iterations. Simply enter the details of sources as you go along, and it will automatically create a perfect bibliography or works cited page at the end. Next to these great beliefs, a biological muzehheroktav.000webhostapp.com theory seems pretty unimportant. The children were often mischievous but always delightful. Finding out background information can help you figure out an angle and identify subjects to interview. A headline should be action-oriented and should convey why the story is important. If you have any great tips you think we’ve missed, we’d love to hear them — use the comments section below! Paul Long posts are harder to proof read. Ramsay That means a lot. Is a personal narrative written in story form? This is also a chance when you can ask a few follow-up questions if you find you need more information. The main task of any writer is to find as many similarities and a difference, as it is possible. Looking at Writing Writing samples from real kids pre-K—3. Jesus was born of the Virgin Mary in Bethlehem, according to the prophecy in Micah 5: The arising of life on earth is indeed a miracle, but we don’t need to base our faith on a flawed mathematical analysis. It guarantees that each payment is secured. Keep under 65 characters long. A newspaper, on the other hand, is meant for a more general audience and may be more open to varied content. My College Guide Magazine – Junior. Fundamentalist Christians will claim that if you say that the first part of Genesis is non-literal, then you’ll say that the entire Bible is non-literal like the proverbial camel’s nose poking in under the edge of the tent. Wow, Ramsay, this is awesome. JB Jana Bautista Jul 17, It means that each paper is written from scratch, following research and investigation of the stated subject. Just two quick notes:. Submitted by Anonymous not verified on February 23, – Technology is to make our lives better, not to worsen them. She started it in and it has gotten bigger and bigger every June About Ron Armstrong: There are a number of ways to write a feature, depending on what you want to focus on. I worked professionally in simulation for 11 years. CoffeeIsLife Little People of America Scholarship: Information you can trust For over 25 years, My College Guide has been producing an annual magazine chock full of expert advice to aid you in your college selection process. And to Mike yes, a sentence that starts with and your logic system allows you to extrapolate that not doing a word search ensures that her child “will never have to face challenges or adversity”. You may ask why God would create an earth that looks older. Are you willing to be a Our Podcasts Watch or listen to our classroom video, author interviews and more. If you wrote the essay on a word processor, you can find out the number of words quickly. She loves the computer and is always wanting to learn how to type faster so this is definatly going to be the way to go. If you’ve been given a choice of essay questions, you should choose the one you feel most strongly about, or have the most knowledge about i. I have investigated the assertions and theories of young-earth creationists. As Christian biologist Kenneth Miller has pointed out, evolution makes it inevitable that somewhere in the universe, some day, there would arise a creature capable of knowing God and loving Him in return. Some young-earth creationists assert that the earth is 10, years old, and others assert that the earth is 6, years old. Just getting the main points down before diving in seems to really help in making an article more focused, and helps to avoid little tangents that detract from the main idea behind the post. Not Helpful 21 Helpful Not only is this scholarship super easy to enter, but you can earn up to three bonus entries when you share it via social media. More importantly, leadership philosophies can change as we grow and adapt in an organization Your posts always teach me something new.

Critical Thinking in Mathematics Education

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• Logical thinking
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• Goals of mathematics education

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How To Encourage Critical Thinking in Math

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Critical thinking is more than just a buzzword… It’s an essential skill that helps students develop problem-solving abilities and make logical connections between different concepts. By encouraging critical thinking in math, students learn to approach problems more thoughtfully, they learn to analyze and evaluate math concepts, identify patterns and relationships, and explore different strategies for finding the solution. Critical thinking also involves a great deal of persistence. Those are critical life skills!

When you think about it, students are typically asked to solve math problems and find the answer. Showing their work is frequently stressed too, which is important, but not the end. Instead, students need to be able to look at math in different ways in order to truly grasp a complete understanding of math concepts. Mathematics requires logical reasoning, problem-solving, and abstract thinking.

What Does Critical Thinking in Math Look Like?

When I think about critical thinking in math, I focus on:

• Solving problems through logical thinking . Students learn how to break down complex problems, analyze the different parts, and understand how they fit together logically.
• Identifying patterns and making connections. Students learn how to identify patterns across different math concepts, make connections between seemingly unrelated topics, and develop a more in-depth understanding of how math works.
• Evaluating and comparing solutions. Students learn to evaluate which solution is best for a given problem and identify any flaws in their reasoning or others’ reasoning when looking at different solutions

Mathematician Posters

These FREE Marvelous Mathematician posters have been a staple in my classroom for the last 8+ years! I first started using a version from MissMathDork and adapted them for my classroom over the years. 

I print, laminate, and add magnetic stickers on the back. At the beginning of the year, I only put one or two up at a time depending on our area of focus. Now, they are all hanging on my board, and I’ll pull out different ones depending on our area of focus. They are so empowering to my mathematicians and help them stay on track!

A Marvelous Mathematician:

• knows that quicker doesn’t mean better
• looks for patterns
• knows mistakes happen and keeps going
• makes sense of the most important details
• embraces challenges and works through frustrations
• uses proper math vocabulary to explain their thinking
• shows their work and models their thinking
• discusses solutions and evaluates reasonableness
• gives context by labeling answers
• applies mathematical knowledge to similar situations
• checks for errors (computational and conceptual)

Critical Thinking Math Activities

Here are a few of my favorite critical thinking activities. 

Square Of Numbers

I love to incorporate challenge problems (use Nrich and Openmiddle to get started) because they teach my students so much more than how to solve a math problem. They learn important lessons in teamwork, persistence, resiliency, and growth mindset. We talk about strategies for tackling difficult problems and the importance of not giving up when things get hard.

This square of numbers challenge was a hit!

ALL kids need to feel and learn to embrace challenge. Oftentimes, kids I see have rarely faced an academic challenge. Things have just come easy to them, so when it doesn’t, they can lack strategies that will help them. In fact, they will often give up before they even get started.

I tell them it’s my job to make sure I’m helping them stretch and grow their brain by giving them challenges. They don’t love it at first, but they eventually do! 

This domino challenge was another one from Nrich . I’m always on the hunt for problems like this!!  How would you guide students toward an answer??

Fifteen Cards

This is a well-loved math puzzle with my students, and it’s amazing for encouraging students to consider all options when solving a math problem.

We have number cards 1-15 (one of each number) and only seven are laid out. With the given clues, students need to figure out which seven cards should be put out and in what order. My students love these, and after they’ve done a few, they enjoy creating their own, too! Use products, differences, and quotients to increase the challenge.

This is also adapted from Nrich, which is an AMAZING resource for math enrichment!

This is one of my favorite fraction lessons that I’ve done for years! Huge shout out to Meg from The Teacher Studio for this one. I give each child a slip of paper with this figure and they have to silently write their answer and justification. Then I tally up the answers and have students take a side and DEBATE with their reasoning! It’s an AMAZING conversation, and I highly recommend trying it with your students. 

Sometimes we leave it hanging overnight and work on visual models to make some proofs. 

Logic Puzzles

Logic puzzles are always a hit too! You can enrich and extend your math lessons with these ‘Math Mystery’ logic puzzles that are the perfect challenge for 4th, 5th, and 6th grades. The puzzles are skills-based, so they integrate well with almost ANY math lesson. You can use them to supplement instruction or challenge your fast-finishers and gifted students… all while encouraging critical thinking about important math skills!

Three levels are included, so they’re perfect to use for differentiation.

• Introductory logic puzzles are great for beginners (4th grade and up!)
• Advanced logic puzzles are great for students needing an extra challenge
• Extra Advanced logic puzzles are perfect for expert solvers… we dare you to figure these puzzles out! 

Do you have a group of students who are ready for more of a fraction challenge? My well-loved fraction puzzlers are absolutely perfect for fraction enrichment. They’ll motivate your students to excel at even the most challenging tasks! 

Math Projects

Math projects are another way to differentiation while building critical thinking skills. Math projects hold so much learning power with their real-world connections, differentiation options, collaborative learning opportunities, and numerous avenues for cross curricular learning too. 

If you’re new to math projects, I shared my best tips and tricks for using math projects in this blog post . They’re perfect for cumulative review, seasonal practice, centers, early finisher work, and more.

I use both concept-based math projects to focus on specific standards and seasonal math projects that integrate several skills.

Error Analysis

Finally, error analysis is always a challenging way to encourage critical thinking. When we use error analysis, we encourage students to analyze their own mistakes to prevent making the same mistakes in the future.

For my gifted students, I use error analysis tasks as an assessment when they have shown mastery of a unit during other tasks. For students in the regular classroom needing enrichment, I usually have them complete the tasks in a center or with a partner.

For students needing extra support, we complete error analysis in small groups.  We go step-by-step through the concept and they are always able to eventually identify what the error is. It is so empowering to students when they finally figure out the error AND it helps prevent them from making the same error in the future!

My FREE addition error analysis is a good place to start, no matter the grade level. I show them the process of walking through the problem and how best to complete an error analysis task.

When you’re ready for more, this bundle of error analysis tasks contains more than 240 tasks to engage and enrich your students in critical thinking practice.

If you want to dig even deeper, visit this conceptual vs computational error analysis post to learn more about using error analysis in the classroom. 

Related Critical Thinking Posts

• How to Increase Critical Thinking and Creativity in Your “Spare” Time
• More Tips to Increase Critical Thinking

Critical thinking is essential for students to develop a deeper understanding of math concepts, problem-solving skills, and a stronger ability to reason logically. When you learn how to encourage critical thinking in math, you’re setting your students up for success not only in more advanced math subjects they’ll encounter, but also in life. 

How do you integrate critical thinking in your classroom? Come share your ideas with us in our FREE Inspired In Upper Elementary Facebook group .

Mary Montero

I’m so glad you are here. I’m a current gifted and talented teacher in a small town in Colorado, and I’ve been in education since 2009. My passion (other than my family and cookies) is for making teachers’ lives easier and classrooms more engaging.

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Mary Thankyou for your inspirational activities. I have just read and loved the morning talk activities. I do have meetings with my students but usually at end of day. What time do you

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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.

Related articles

Developing Critical Numeracy Across the Curriculum

• Critical Numeracy
• Teaching Activities
• Thinking Strategies
• Student Reasoning
• Teacher Experiences
• giving students an opportunity to make sense of the mathematical concepts and make sense of the context before moving to more critical thinking about both.
• giving an opportunity for exploration with others - pairs, groups or whole class discussions where different views are juxtaposed and reconciled.
• giving an opportunity for students to create something using their new knowledge - particularly products which have an audience greater themselves, enabling feedback from a wider community.

Critical Numeracy = understanding + criticality + community + creativity

The Four Resource Model for Critical Numeracy

Critical Numeracy uses a similar model to the Four Resource Model of Critical Literacy (Luke and Freebody ) to build students' capacities to ask questions about the meaning, validity and usefulness of texts containing mathematical concepts or information. By using a similar model to critical literacy students can recruit and build on the visible thinking strategies that they are developing whether in a literacy or numeracy context.

In the process of  applying a critical numeracy lens , students go deeper into the mathematical ideas and deeper into the contexts. They challenge the usefulness of the mathematical ideas in relationship to the context. So rather than just learning the rules of maths they are encouraged to explore and question their application.

The following gives examples of the type of questions you may ask to help students become familiar with the mathematical ideas and the contexts ( De-coding and Meaning-Making ) before applying a more critical lens ( Using and Analysing ). We also give examples of how to help the students creatively use their understandings. We expand on this model in Thinking Strategies where we link to possible Thinking routines that can assist in developing student capacity in each of the quadrants.

Printer friendly version of the Four Resource Model for Critical Numeracy

Website updated 2009

Encouraging Early Numeracy Skills in Your Program

• June 28, 2016

Early numeracy skills, like early literacy, are an important component of school readiness efforts and are an everyday part of even very young children’s lives.

Although definitions of numeracy vary, it is typically understood to mean “the ability to use appropriate mathematical knowledge, …skills,… and experience whenever they are needed in everyday life” (Department of Education of the Arts in Perry, 11). Early numeracy skills, like early literacy, are an important component of school readiness efforts and are an everyday part of even very young children’s lives.

Early mathematical concepts and skills—those which the first-grade mathematics curriculum builds upon—include those listed below (Bowman, B. T., Donovan, M. S., & Burns, M. S., (Eds.), 2001, 76).

• Recognition of size, shape, and patterns
• Ability to count verbally (first forward, then backward)
• Recognition of numerals
• Ability to identify more and less of a quantity
• Mastery of one-to-one correspondence (i.e., matching sets, or knowing which group has four and which has five)

To achieve these skills requires competency in the four keystones of mathematical literacy listed below (Diezmann and Yelland, 2000, pp. 49-52):

• Representation —finding ways to express mathematical concepts with words, diagrams, pictures, symbols and manipulatives (like blocks) Casey (aged 3) was making piles of sunflower seeds. When asked why, he explained: “We need 5 seeds for each kid. I’m helping!”
• Performing Mathematical Manipulations— utilizing the appropriate calculations and procedures, using the information given to answer the question posed Yvette (28 months) and Charlotte (30 months) were standing in the center’s outside play area and crying. When their caregiver, Julia, came over to see why, Yvette explained: “No trikes!” Julia watched several children zooming around on tricycles. “Hmmmm, we need more trikes, you’re right. How many more?” Yvette shouted, “One for me and one for Charlie!” Julia said, “That’s right. We need 2 trikes for you and Charlie. Let’s see if anyone is done riding them yet.”
• Reasoning— assessing a problem using facts, properties and relationships to make and test conjectures, develop logical arguments, and identify a useful answer. Carl (aged 9 months) looked at the toy his early interventionist brought that day. It was a plastic drum with 3 holes in the top. The holes were in the shape of a triangle, a circle and a square. Carl looked at the blocks surrounding him. They, too, were different shapes. Carl picked up a triangular block. He put it in his month, then banged it on the floor. He touched the edges with his fingers. Then he tried to stuff it in each of the holes of the new toy. Surprise! It fell inside the triangle hole! Carl reached for another block, a circular one this time…
• Problem solving— thinking through an issue, using prior knowledge and skills to reach and validate a solution (and realizing more than one solution may exist). Posing one’s own problems. Jennie (aged 18 months) was given a bag of crackers. “Give everybody one,” her caregiver, Teri, said. Jennie gave each child at the table a cracker. But there was one left over. “What should we do with the extra cracker?” asked Teri. Jennie stopped and thought. Then she opened her mouth and popped it in!

Here, numeracy, language and social skills come together. In all of the competencies above, children’s ability to communicate effectively and express their ideas is integral to success with a given mathematical problem—whether it comes up in play (only five children may be in the blocks area at once) or in a more structured setting. Early numeracy experiences provide ample opportunities to develop and extend children’s vocabulary and literacy skills.

Even very young children’s social environments include access to counting systems (how many steps are we walking up?), as well as exposure to principles of one-to-one correspondence (are there enough cookies for each child?) (Bowman, B. T., et al., 2001, p. 201). Prior to entering school, most children develop an innate understanding of addition and subtraction through everyday interactions (Thomas has two shovels; Joseph wants one; after this transaction, Thomas sees that he has one left) (Bowman B. T. et al., 2001, p. 201). This learning occurs at home as well, where parents introduce important numerical concepts through everyday interactions. For example, young children learn that money has a value when they are denied a toy deemed “too expensive.” Family members and other adults support children’s number learning through songs like “One Two, Buckle My Shoe” and stories like “The Three Bears.” This informal foundation in mathematics, forged during very young children’s prior-to-school years, provides a rudimentary conceptual framework from which formal instruction can begin.

The tips below highlight ways that staff members can build upon the natural curiosity of very young children to establish a solid foundation of early numeracy skills. (Note: Most of these tips are designed for older children—ages 2-3. Younger children can be exposed to stories and songs using repetition, rhymes and numbers. Staff, parents and caregivers should also speak to children about mathematical concepts (e.g., “That’s a great big dog! He’s bigger than our cat, isn’t he?”) even though babies don’t yet understand.

Promoting Early Numeracy Skills in Infant/Family Programs

• Help children learn their own address and phone number. Talk with them about how buildings are numbered, how their house or apartment is one of a series, each with its own number.
• Use dress-up time as an opportunity to talk about relative sizes (“the red dress is bigger than the green dress”) as well as to help children begin to think about their own size relative to other objects (“will that policeman’s jacket fit you, do you think?”).
• Home visitors (and center-based staff) can use cooking as a way to introduce mathematical concepts. Even young children can be helped to fill, stir, and pour—through these activities, children learn to count, measure, divide, and estimate.
• Taking a walk gives children many opportunities to compare (“which leaf is bigger?”), assess (“how many leaves does that branch have?”), note similarities and differences (“does the bunny live in the tree? Does the bird?”) and categorize (“see if you can find all the red leaves”).
• Use an hourglass or timer to time activities. This helps children develop a sense of time and to understand that some things take longer than others.
• Point out the different shapes and colors you see during the day. On a walk, you may see a triangle-shaped sign that’s yellow. Inside you may see a rectangle-shaped sign that’s red.
• Sing songs that rhyme, repeat, or have numbers in them. Songs reinforce patterns (which are tools that help in problem-solving because they tell us what to expect next). They also are fun ways to practice language and mathematics.
• Use a calendar. Talk to children about what day it is, what the different days of the week are, and what they can expect the next day.
• Ask for children’s help in distributing items to class members. Requests like “Give one to each child” helps children understand one-to-one correspondence. When you are distributing items, emphasize the number concept: “One for you, one for me, one for Raul.”
• Ask children to sort crayons and markers into containers (or use buttons, pasta, leaves, shells, etc.).
• Provide children with lots of open-ended play with blocks, empty boxes, milk cartons, etc. Stacking and manipulating these toys help children to learn about shapes and the relationships among shapes (e.g., two triangles make a square). Nesting boxes for younger children help them to understand the relationships between different sized objects.
• Give children a cardboard box or tunnel to climb through. This helps them understand their body in space (this is called “spatial relations”).
• Hang pieces of yarn and ask which is the longest and shortest. Have kids arrange the yarn from longest to shortest.
• Pass a shape around and have children look at it and feel it with eyes open and closed.
• Help children practice patterns by stringing beads, making block patterns, or making a pattern with two napkins at snack time.

References:

• Bowman, B. T., Donovan, M. S., & Burns, M. S., (Eds.). (2001).  Eager to learn: Educating our preschoolers.  Washington, DC: National Academy of Sciences.
• Diezmann, C., & Yelland, N. J. (2000). Developing mathematical literacy in the early childhood years. In Yelland, N. J. (Ed.),  Promoting meaningful learning: Innovations in educating early childhood professionals.  (pp.47-58). Washington, DC: National Association for the Education of Young Children.
• Perry, B. (1998/1999).  Early childhood numeracy.  Canberra: Commonwealth of Australia. Available at:  http://www.aamt.edu.au/NUMERACY/PERRY.PDF

Internet Resources:

• www.naeyc.org/resources/eyly/1997/21.htm
• http://members.tripod.com/~Patricia_F/mathscience.html
• www.ed.gov/pubs/EarlyMath/title.html

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