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How To Encourage Critical Thinking in Math

By Mary Montero

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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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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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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

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!

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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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Re-highlighting the potential of critical numeracy

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• Published: 09 December 2019
• Volume 33 , pages 285–299, ( 2021 )

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• Kathyrn Paige   ORCID: orcid.org/0000-0001-9238-0765 1  

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Since the development of the term numeracy in the mid-fifties in the United Kingdom, the concept of what numeracy is and how it is defined has undergone many changes. The Australian education community continues to explore the concept of numeracy, consider the place and definition of numeracy in the curriculum and consider its relationship with mathematics. Currently, the Australian curriculum incorporates mathematics as one of seven learning areas and numeracy as one of seven general capabilities. Numeracy is defined as the capacity, confidence and disposition to use mathematics. Other general capabilities include critical and creative thinking, ethical behaviour, personal and social competence and intercultural understanding. This paper explores numeracy from an Australian Curriculum perspective and extends this view to consider the potential of “critical numeracy”. Using the lens of Freire’s critical pedagogy, this paper discusses a perspective of critical pedagogy for mathematics and numeracy that challenges teachers to address student lifeworlds, their community and social issues.

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This paper stems from the authors’ work as part of a symposium entitled Just Pedagogy presented at the annual conference of the Australian Association for Research in Education in 2016 (O’Keeffe and Paige 2016 ).

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O’Keeffe, L., Paige, K. Re-highlighting the potential of critical numeracy. Math Ed Res J 33 , 285–299 (2021). https://doi.org/10.1007/s13394-019-00297-8

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5 Ways to Stop Thinking for Your Students

Too often math students lean on teachers to think for them, but there are some simple ways to guide them to think for themselves.

Who is doing the thinking in your classroom? If you asked me that question a few years ago, I would have replied, “My kids are doing the thinking, of course!” But I was wrong. As I reflect back to my teaching style before I read Building Thinking Classrooms by Peter Liljedahl (an era in my career I like to call “pre-thinking classroom”), I now see that I was encouraging my students to mimic rather than think .

My lessons followed a formula that I knew from my own school experience as a student and what I had learned in college as a pre-service teacher. It looked like this: Students faced me stationed at the board; I demonstrated a few problems while students copied what I wrote in their notes. I would throw out a few questions to the class to assess understanding. If a few kids answered correctly, I felt confident that the lesson had gone well. Some educators might call this “ I do, we do, you do .”

What’s wrong with this formula? When it was time for them to work independently, which usually meant a homework assignment because I used most of class time for direct instruction, the students would come back to class and say, “The homework was so hard. I don’t get it. Can you go over questions 1–20?” Exhausted and frustrated, I would wonder, “But I taught it—why didn’t they get it?”

Now in the “peri-thinking classroom” era of my career, my students are often working at the whiteboards in random groups as outlined in Liljedahl’s book. The pendulum has shifted from the teacher doing the thinking to the students doing the thinking. Do they still say, “I don’t get it!”? Yes, of course! But I use the following strategies to put the thinking back onto them.

5 Ways to Get Your Students to Think

1. Answer questions with a refocus on the students’ point of view. Liljedahl found in his research that students ask three types of questions: “(1) proximity questions—asked when the teacher is close; (2) stop thinking questions—most often of the form ‘is this right’ or ‘will this be on the test’; and (3) keep thinking questions—questions that students ask so they can get back to work.” He suggests that teachers acknowledge “proximity” and “stop thinking questions” but not answer them.

Try these responses to questions that students ask to keep working:

• “What have you done so far?” 
• “Where did you get that number?” 
• “What information is given in the problem?” 
• “Does that number seem reasonable in this situation?”  

2. Don’t carry a pencil or marker. This is a hard rule to follow; however, if you hold the writing utensil, you’ll be tempted to write for them . Use verbal nudges and hints, but avoid writing out an explanation. If you need to refer to a visual, find a group that has worked out the problem, and point out their steps. Hearing and viewing other students’ work is more powerful .

3. We instead of I . When I assign a handful of problems for groups to work on at the whiteboards, they are tempted to divvy up the task. “You do #30, and I’ll do #31.” This becomes an issue when they get stuck. I inevitably hear, “Can you help me with #30? I forgot how to start.”

I now require questions to use “we” instead of “I.” This works wonders. As soon as they start to ask a question with “I,” they pause and ask their group mates. Then they can legitimately say, “ We tried #30, and we are stumped.” But, in reality, once they loop in their group mates, the struggling student becomes unstuck, and everyone in the group has to engage with the problem.

4. Stall your answer. If I hear a basic computation question such as, “What is 3 divided by 5?” I act like I am busy helping another student: “Hold on, I need to help Marisela. I’ll be right back.” By the time I return to them, they are way past their question. They will ask a classmate, work it out, or look it up. If the teacher is not available to think for them, they learn to find alternative resources.

5. Set boundaries. As mentioned before, students ask “proximity” questions because I am close to them. I might reply with “Are you asking me a thinking question? I’m glad to give you a hint or nudge, but I cannot take away your opportunity to think.” This type of response acknowledges that you are there to help them but not to do their thinking for them.

When you set boundaries of what questions will be answered, the students begin to more carefully craft their questions. At this point of the year, I am starting to hear questions such as, “We have tried solving this system by substitution, but we are getting an unreasonable solution. Can you look at our steps?” Yes!

Shifting the focus to students doing the thinking not only enhances their learning but can also have the effect of less frustration and fatigue for the teacher. As the class becomes student-centered, the teacher role shifts to guide or facilitator and away from “sage on the stage.”

As another added benefit, when you serve as guide or facilitator, the students are getting differentiated instruction and assessment. Maybe only a few students need assistance with adding fractions, while a few students need assistance on an entirely different concept. At first, you might feel like your head is spinning trying to address so many different requests; however, as you carefully sift through the types of questions you hear, you will soon be comfortable only answering the “keep thinking” questions.

Critical connections between Numeracy and Mathematics

Increasingly research is showing that life and work in in the 21st century is requiring higher levels of mathematics and numeracy of its citizens. This paper looks at the implications of this for the skills we want our students to develop and leave school with, and how we can better address these in our teaching and learning.

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Why connections between numeracy and mathematics is an important issue in mathematics teaching and learning in the 21st century.

Increasingly research is showing that life and work in in the 21st century is requiring higher levels of mathematics and numeracy of its citizens. Numeracy and mathematics are intrinsically connected and BOTH are needed in our ever changing, globalised and technological world. This paper looks at the implications of this for the skills we want our students to develop and leave school with, and how we can better address these in our teaching and learning.

Research about 21st century skills

Research is showing that the skills and knowledge now needed to succeed in work, life and citizenship have significantly changed in the 21st century, often driven by technological advances and an ever-increasing use of numerical and quantitative information and data. This also connects with the transforming nature of the workforce associated with Industry 4.0 and the Gig economy, with increasing demands for science, technology, engineering and mathematics (STEM) skills (e.g. see AAMT and AiGroup 2014; Binkley et al. 2012; FYA 2017; Gravemeijer et al 2017; Griffin et al. 2012; Hoyles et al. 2010; NCTM, 2017; P21 2016; Pellegrino et al. 2012; Wake 2015).

In their 2017 review of mathematics education for the 21st century, the National Council of Teachers of Mathematics (NCTM) in the USA (2017) argued that mathematics is at the heart of most innovations in the information economy. They saw mathematical and statistical literacy as needed more than ever to filter, understand, and act on the enormous amount of data and information that we encounter every day.

What numeracy and maths skills?

One of the key outcomes of such research is that the mathematics, or numeracy, related tasks that people undertake involve much more than basic arithmetic skills and straightforward procedural competence. For example, in an Australian project undertaken by practising maths teachers for the Australian Association of Mathematics Teachers (AAMT) and funded by the Office of the Chief Scientist, Quantitative Skills in 21st Century Workplaces, the research identified and analysed the gaps between young peoples' numeracy skills and the expectations of 21st century workplaces. Mathematics was considered extremely important in all of the companies involved, and changing work practices were found to be generating new demands for mathematical skills, particularly in areas such as efficiency, innovation and Quality & Continuous Improvement.

The application of mathematics in the workplace is not straightforward and goes well beyond a command of ‘core’ mathematical content. Workers perform sophisticated functions which require them to be confident to use mathematical skills in problem-solving situations and to see the consequences of the mathematics related procedures. (AAMT & AiGroup, 2014)

The project found that workers needed a blend of the following key skills:

• ability to recognise and identify how and when mathematics is used in the workplace
• an understanding of mathematical concepts, procedures and skills
• an understanding of the kinds of practical tasks they need to perform
• the strategic processes they should be able to use in using and applying mathematics.

Although the skills observed appeared to be fundamental, it is their use and application in work contexts that is not straightforward (AAMT & AiGroup, 2014).

The critical connections between numeracy and mathematics

What is meant by the term numeracy, and what is its relationship to terms such as quantitative or mathematical literacy, and how does it connect to the world of (school) mathematics?

The term numeracy is used in some countries, like in Australia, however, other expressions are used as well, for example, mathematical literacy or quantitative literacy. This is further complicated by the lack of an equivalent term in some languages. Moreover, what is meant by numeracy also varies between countries, and can vary between how it is understood when applied to school education compared to within adult education. However, increasingly numeracy now refers to the capability to use a range of mathematical and statistical knowledge and skills to solve problems in the real world for a purpose.

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)

One underpinning connection and difference that needs to be acknowledged and highlighted is between numeracy and mathematics as a domain of knowledge. The meaning of numeracy used here encompasses the need for individuals to be able to understand, use and apply mathematical (and statistical) skills and knowledge.

Thus, to be considered numerate, it is expected that people will need to know some mathematics, and be able to apply that mathematics within a real-world context.

Why this matters

With 21st-century life and workplaces requiring more critical, reflective mathematical reasoning skills and the ability to interpret and understand a broader range of data and processes, our school leavers need better numeracy and maths skills than ever before. 

Hence schools need to teach both numeracy and mathematics well—within maths classes by maths teachers and also as part of numeracy across the curriculum.

Critical element to numeracy

The issue of the need to be critical, and use reflective mathematical reasoning skills is now identified as a key skill and is routinely included in many numeracy frameworks as will be shown below. An emphasis on critical reflection or interpretation has mainly arisen out of adult numeracy research and development, where, for example, Johnston (1994) argued that

To be numerate is more than being able to manipulate numbers, or even being able to 'succeed' in school or university mathematics. Numeracy is a critical awareness which builds bridges between mathematics and the real-world, with all its diversity (p. 34).

Key terms and definitions

The above views about the need for numeracy and maths skills is recognised in Australian curriculum. For example, the Australian Curriculum: Mathematics (V 8.4) includes numeracy as one of the General Capabilities that must be observed while teaching and learning mathematics stating:

… students become numerate as they develop the knowledge and skills to use mathematics confidently across other learning areas at school and in their lives more broadly. Numeracy encompasses the knowledge, skills, behaviours and dispositions that students need to use mathematics in a wide range of situations. It involves students recognising and understanding the role of mathematics in the world and having the dispositions and capacities to use mathematical knowledge and skills purposefully.

Similarly, the Victorian Curriculum and Assessment Authority defines numeracy as:

Numeracy is the knowledge, skills, behaviours and dispositions that students need in order to use mathematics in a wide range of situations.

Internationally, the related term mathematical literacy is used in the OECD's Programme for International Student Assessment (PISA). It is defined as:

… an individual's capacity to formulate, employ, and interpret mathematics in a variety of contexts. It includes reasoning mathematically and using mathematical concepts, procedures, facts, and tools to describe, explain, and predict phenomena. It assists individuals to recognise the role that mathematics plays in the world and to make the well-founded judgments and decisions needed by constructive, engaged and reflective citizens. (OECD, 2019, p. 75)

It should be noted that at the conceptual and implementation level, numeracy and mathematical literacy are very closely related constructs in terms of their core, underlying ideas (e.g., see Gal & Tout, 2014).

As with the above definitions, most 21 st century descriptions and definitions of numeracy now refer to the need to use and apply a range of mathematical knowledge.

It is important to note therefore that numeracy is not seen as synonymous with a minimal or low level of mathematical knowledge and skills—but covers a broad spectrum of skills and knowledge. Sometimes numeracy can be viewed as a lower level skill, simply being about numbers and arithmetical operations.

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20 Math Critical Thinking Questions to Ask in Class Tomorrow

• November 20, 2023

The level of apathy towards math is only increasing as each year passes and it’s up to us as teachers to make math class more meaningful . This list of math critical thinking questions will give you a quick starting point for getting your students to think deeper about any concept or problem. 

Since artificial intelligence has basically changed schooling as we once knew it, I’ve seen a lot of districts and teachers looking for ways to lean into AI rather than run from it.

The idea of memorizing formulas and regurgitating information for a test is becoming more obsolete. We can now teach our students how to use their resources to make educated decisions and solve more complex problems.

With that in mind, teachers have more opportunities to get their students thinking about the why rather than the how.

Table of Contents

Looking for more about critical thinking skills? Check out these blog posts:

• Why You Need to Be Teaching Writing in Math Class Today
• How to Teach Problem Solving for Mathematics
• Turn the Bloom’s Taxonomy Verbs into Engaging Math Activities

What skills do we actually want to teach our students?

As professionals, we talk a lot about transferable skills that can be valuable in multiple jobs, such as leadership, event planning, or effective communication. The same can be said for high school students. 

It’s important to think about the skills that we want them to have before they are catapulted into the adult world. 

Do you want them to be able to collaborate and communicate effectively with their peers? Maybe you would prefer that they can articulate their thoughts in a way that makes sense to someone who knows nothing about the topic.

Whatever you decide are the most essential skills your students should learn, make sure to add them into your lesson objectives.

When should I ask these math critical thinking questions?

Critical thinking doesn’t have to be complex or fill an entire lesson. There are simple ways that you can start adding these types of questions into your lessons daily!

Start small

Add specific math critical thinking questions to your warm up or exit ticket routine. This is a great way to start or end your class because your students will be able to quickly show you what they understand. 

Asking deeper questions at the beginning of your class can end up leading to really great discussions and get your students talking about math.

Add critical thinking questions to word problems

Word problems and real-life applications are the perfect place to add in critical thinking questions. Real-world applications offer a more choose-your-own-adventure style assignment where your students can expand on their thought processes. 

They also allow your students to get creative and think outside of the box. These problem-solving skills play a critical role in helping your students develop critical thinking abilities.

Keep reading for math critical thinking questions that can be applied to any subject or topic!

When you want your students to defend their answers.

• Explain the steps you took to solve this problem
• How do you know that your answer is correct?
• Draw a diagram to prove your solution.
• Is there a different way to solve this problem besides the one you used?
• How would you explain _______________ to a student in the grade below you?
• Why does this strategy work?
• Use evidence from the problem/data to defend your answer in complete sentences.

When you want your students to justify their opinions

• What do you think will happen when ______?
• Do you agree/disagree with _______?
• What are the similarities and differences between ________ and __________?
• What suggestions would you give to this student?
• What is the most efficient way to solve this problem?
• How did you decide on your first step for solving this problem?

When you want your students to think outside of the box

• How can ______________ be used in the real world?
• What might be a common error that a student could make when solving this problem?
• How is _____________ topic similar to _______________ (previous topic)?
• What examples can you think of that would not work with this problem solving method?
• What would happen if __________ changed?
• Create your own problem that would give a solution of ______________.
• What other math skills did you need to use to solve this problem?

Let’s Recap:

• Rather than running from AI, help your students use it as a tool to expand their thinking.
• Identify a few transferable skills that you want your students to learn and make a goal for how you can help them develop these skills.
• Add critical thinking questions to your daily warm ups or exit tickets.
• Ask your students to explain their thinking when solving a word problem.
• Get a free sample of my Algebra 1 critical thinking questions ↓

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Academic literacy is more than language, it’s about critical thinking and analysis: universities should do more to teach these skills

Associate professor, University of Pretoria

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Pineteh Angu is a member of the English Academy of South Africa (EASA), European Association for the Teaching of Academic Writing (EATAW) and the newly formed South African Association of Academic Literacy Practitioners (SAAALP)

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Making the adjustment from school to university is no easy task. For instance, there’s a big difference between writing a high school essay and crafting an academic paper which meets university standards.

In the decades since formal apartheid ended, South Africa’s universities have become increasingly accessible to students from different socioeconomic, schooling and linguistic backgrounds. But many of these students do not have the language or literacy skills to succeed at university level.

When I talk about “language”, I don’t mean that their level of fluency in English is the problem. In my long experience as a researcher and practitioner in the field of academic literacy, I have seen time and again that not only non-native English speakers struggle to transition from school to university. Many students, no matter what language they speak, lack the skills of critical thinking, analysis and logical reasoning.

Academic literacy is a mode of reasoning that aims to develop university students into deep thinkers, critical readers and writers. Many universities in South Africa offer academic literacy programmes to support struggling undergraduates. On paper, these programmes are an opportunity for students to read and analyse different academic texts. Ideally they should provide students with the academic tools to cope in an ever-changing university landscape and the broader South African economy.

But, as my research and that of other academic literacy practitioners shows , many South African universities’ academic literacy programmes are still promoting what researchers in this field call a “ deficit model ”. Here, lecturers assume that academic literacy is about teaching generic skills that can be transferred across disciplines. These skills include note-taking, structuring an academic essay and constructing sentences and paragraphs. There’s also a big focus on the rules of English grammar.

While these are all useful skills, academic literacy is about so much more.

This approach does not equip students with skills that can transform their minds : critical and logical reasoning, argumentation, conceptual and analytical thinking, and problem solving.

Without these skills, undergraduate students come to believe, for instance, that disciplinary knowledge is factual and truthful and cannot be challenged. They don’t learn how to critically assess and even challenge knowledge. Or they only see certain forms of knowledge as valid and scientific. In addition, they believe that some (mainly African) languages can never be used for research, teaching and learning. Pragmatically, they also don’t develop the confidence to notice their own errors, attempt to address them or seek help.

I would like to share some suggestions on how to produce university graduates who can think critically.

The deficit model

Why does the deficit model still prevail in South African universities? Research ( including mine ) offers some clues.

First, academic literacy still suffers from confusion around the definition. Not everyone in higher education agrees on what it is. So, disciplinary experts and some academic literacy practitioners misrepresent it as English language support. They assume that reading and writing in English with grammatical correctness is more important than critical thinking and argumentation.

They assume that a semester or year-long academic literacy course can “fix” students who lack these basic English skills. This approach tends to target and stigmatise people whose home language isn’t English, most often Black South Africans, Afrikaans speakers and students from other parts of Africa.

Another issue is that some academic literacy lecturers are not familiar with or are unconcerned about new research. They don’t follow national or global scholarly debates about the discipline. That means their teaching isn’t grounded in research or in new theoretical shifts.

Moreover, academic literacy practitioners and disciplinary experts do not always work together to develop the courses. This entrenches misleading views about the field, and it means academic literacy lecturers are not always aware of what’s expected in different disciplines.

Doing things differently

These problems can be overcome.

Academic literacy programmes at South African universities should focus on providing students with empowering academic literacy skills that can transform their minds.

The starting point is to understand that academic literacy is a cognitive process. It helps students to think, read and write critically.

For this to happen, disciplinary boundaries and hierarchies must be disrupted. Academic literacy programmes should be designed collaboratively with disciplinary experts . This will guarantee contextual relevance. Academic literacy departments or units need to be staffed by academics who keep abreast of new research in the field. They should be familiar especially with research that focuses on the South African context.

• Critical thinking
• South African universities
• English langauge
• African languages
• Academic writing
• Undergraduates
• Language of instruction
• South African languages
• Study skills

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How Do You Counter Misinformation? Critical Thinking Is Step One

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Late last year, in the days before the Slovakian parliamentary elections, two viral audio clips threatened to derail the campaign of a pro-Western, liberal party leader named Michal Šimečka. The first was a clip of Šimečka announcing he wanted to double the price of beer, which, in a nation known for its love of lagers and pilsners, is not exactly a popular policy position.

In a second clip, Šimečka can be heard telling a journalist about his intentions to commit fraud and rig the election. Talk about career suicide, especially for someone known as a champion of liberal democracy.

There was, however, just one issue with these audio clips: They were completely fake.

The International Press Institute has called this episode in Slovakia the first time that AI deepfakes — fake audio clips, images, or videos generated by artificial intelligence — have played a prominent role in a national election. While it’s unclear whether these bogus audio clips were decisive in Slovakia’s electoral contest, the fact is Šimečka’s party lost the election, and a pro-Kremlin populist now leads Slovakia.

In January, a report from the World Economic Forum found that over 1,400 security experts consider misinformation and disinformation (misinformation created with the intention to mislead) the biggest global risk in the next two years — more dangerous than war, extreme weather events, inflation, and everything else that’s scary. There are a bevy of new books and a constant stream of articles that wrestle with this issue. Now even economists are working to figure out how to fight misinformation.

In a new study , “Toward an Understanding of the Economics of Misinformation: Evidence from a Demand Side Field Experiment on Critical Thinking,” economists John A. List, Lina M. Ramírez, Julia Seither, Jaime Unda and Beatriz Vallejo conduct a real-world experiment to see whether simple, low-cost nudges can be effective in helping consumers to reject misinformation. (Side note: List is a groundbreaking empirical economist at the University of Chicago, and he’s a longtime friend of the show and this newsletter ).

While most studies have focused on the supply side of misinformation — social media platforms, nefarious suppliers of lies and hoaxes, and so on — these authors say much less attention has been paid to the demand side: increasing our capacity, as individuals, to identify and think critically about the bogus information that we may encounter in our daily lives.

A Real-Life Experiment To Fight Misinformation

The economists conducted their field experiment in the run-up to the 2022 presidential election in Colombia. Like the United States, Colombia is grappling with political polarization. Within a context of extreme tribalism, the authors suggest, truth becomes more disposable and the demand for misinformation rises. People become willing to believe and share anything in their quest for their political tribe to win.

To figure out effective ways to lower the demand for misinformation, the economists recruited over 2,000 Colombians to participate in an online experiment. These participants were randomly distributed into four different groups.

One group was shown a video demonstrating “how automatic thinking and misperceptions can affect our everyday lives.” The video shows an interaction between two people from politically antagonistic social groups who, before interacting, express negative stereotypes about the other’s group. The video shows a convincing journey of these two people overcoming their differences. Ultimately, they express regret over unthinkingly using stereotypes to dehumanize one another. The video ends by encouraging viewers to question their own biases by “slowing down” their thinking and thinking more critically.

Another group completed a “a personality test that shows them their cognitive traits and how this makes them prone to behavioral biases.” The basic idea is they see their biases in action and become more self-aware and critical of them, thereby decreasing their demand for misinformation.

A third group both watched the video and took the personality test.

Finally, there was a control group, which neither watched the video nor took the personality test.

To gauge whether these nudges get participants to be more critical of misinformation, each group was shown a series of headlines, some completely fake and some real. Some of these headlines leaned left, others leaned right, and some were politically neutral. The participants were then asked to determine whether these headlines were fake. In addition, the participants were shown two untrue tweets, one political and one not. They were asked whether they were truthful and whether they would report either to social media moderators as misinformation.

What They Found

The economists find that the simple intervention of showing a short video of people from politically antagonistic backgrounds getting along inspires viewers to be more skeptical of and less susceptible to misinformation. They find that participants who watch the video are over 30 percent less likely to “consider fake news reliable.” At the same time, the video did little to encourage viewers to report fake tweets as misinformation.

Meanwhile, the researchers find that the personality test, which forces participants to confront their own biases, has little or no effect on their propensity to believe or reject fake news. It turns out being called out on our lizard brain tribalism and other biases doesn’t necessarily improve our thinking.

In a concerning twist, the economists found that participants who both took the test and watched the video became so skeptical that they were about 31 percent less likely to view true headlines as reliable. In other words, they became so distrustful that even the truth became suspect. As has become increasingly clear, this is a danger in the new world of deepfakes: not only do they make people believe untrue things, they also may make people so disoriented that they don’t believe true things.

As for why the videos are successful in helping to fight misinformation, the researchers suggest that it’s because they encourage people to stop dehumanizing their political opponents, think more critically, and be less willing to accept bogus narratives even when it bolsters their political beliefs or goals. Often — in a sort of kumbaya way — centrist political leaders encourage us to recognize our commonalities as fellow countrymen and work together across partisan lines. It turns out that may also help us sharpen our thinking skills and improve our ability to recognize and reject misinformation.

Critical Thinking In The Age Of AI

Of course, this study was conducted back in 2022. Back then, misinformation, for the most part, was pretty low-tech. Misinformation may now be getting turbocharged with the rapid proliferation and advancement of artificial intelligence.

List and his colleagues are far from the first scholars to suggest that helping us become more critical thinkers is an effective way to combat misinformation. University of Cambridge psychologist Sander van der Linden has done a lot of work in the realm of what’s known as “psychological inoculation,” basically getting people to recognize how and why we’re susceptible to misinformation as a way to make us less likely to believe it when we encounter it. He’s the author of a new book called Foolproof: Why Misinformation Infects Our Minds and How to Build Immunity . Drawing an analogy to how vaccinations work, Van der Linden advocates exposing people to misinformation and showing how it’s false as a way to help them spot and to reject misinformation in the wild. He calls it “prebunking” (as in debunking something before it happens).

Of course, especially with the advent of AI deepfakes, misinformation cannot only be combated on the demand side. Social media platforms, AI companies, and the government will all likely have to play an important role. There’s clearly a long way to go to overcoming this problem, but we have recently seen some progress. For example, OpenAI recently began “watermarking” AI-generated images that their software produces to help people spot pictures that aren’t real. And the federal government recently encouraged four companies to create new technologies to help people distinguish between authentic human speech and AI deepfakes.

This new world where the truth is harder to believe may be pretty scary. But, as this new study suggests, nudges and incentives to get us to slow our thinking, think more critically, and be less tribal could be an important part of the solution.

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Welcome to Grade 7 Mathematics Curriculum Guide! This comprehensive syllabus is tailored to meet the academic needs of students in level 7. Our curriculum is designed to provide a structured approach to learning mathematical concepts, fostering critical thinking, problem-solving skills, and a deep understanding of mathematical principles. Each topic is carefully crafted to align with national standards, ensuring students are well-prepared for future academic challenges. Explore the syllabus below to discover the exciting journey awaiting you in Grade 7 mathematics!

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Grade 7 Maths curriculum builds upon foundational concepts, introducing algebra, geometry, measurement, and data analysis. Students deepen their understanding of numbers, equations, shapes, and problem-solving, fostering critical thinking and real-world application skills.

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Ans: Encourage regular practice, provide access to supplementary resources, and engage in math-related activities together. Communicate with your child’s teacher to understand specific areas where your child may need extra support.

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Ans: Grade 7 mathematics forms a foundation for more advanced mathematical concepts in later grades and is essential for developing critical thinking and problem-solving skills applicable in various real-life situations and future academic pursuits.

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Ans: Teachers can use this guide to plan lessons, assessments, and activities that align with the curriculum standards. They can also customize instruction to meet the needs and interests of their students while ensuring mastery of essential mathematical concepts.

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How do you counter misinformation? Critical thinking is step one

Greg Rosalsky

Late last year, in the days before the Slovakian parliamentary elections, two viral audio clips threatened to derail the campaign of a pro-Western, liberal party leader named Michal Šimečka. The first was a clip of Šimečka announcing he wanted to double the price of beer, which, in a nation known for its love of lagers and pilsners, is not exactly a popular policy position.

In a second clip, Šimečka can be heard telling a journalist about his intentions to commit fraud and rig the election. Talk about career suicide, especially for someone known as a champion of liberal democracy.

There was, however, just one issue with these audio clips: They were completely fake.

The International Press Institute has called this episode in Slovakia the first time that AI deepfakes — fake audio clips, images, or videos generated by artificial intelligence — have played a prominent role in a national election. While it's unclear whether these bogus audio clips were decisive in Slovakia's electoral contest, the fact is Šimečka's party lost the election, and a pro-Kremlin populist now leads Slovakia.

In January, a report from the World Economic Forum found that over 1,400 security experts consider misinformation and disinformation (misinformation created with the intention to mislead) the biggest global risk in the next two years — more dangerous than war, extreme weather events, inflation, and everything else that's scary. There are a bevy of new books and a constant stream of articles that wrestle with this issue. Now even economists are working to figure out how to fight misinformation.

In a new study , "Toward an Understanding of the Economics of Misinformation: Evidence from a Demand Side Field Experiment on Critical Thinking," economists John A. List, Lina M. Ramírez, Julia Seither, Jaime Unda and Beatriz Vallejo conduct a real-world experiment to see whether simple, low-cost nudges can be effective in helping consumers to reject misinformation. (Side note: List is a groundbreaking empirical economist at the University of Chicago, and he's a longtime friend of the show and this newsletter ).

While most studies have focused on the supply side of misinformation — social media platforms, nefarious suppliers of lies and hoaxes, and so on — these authors say much less attention has been paid to the demand side: increasing our capacity, as individuals, to identify and think critically about the bogus information that we may encounter in our daily lives.

A Real-Life Experiment To Fight Misinformation

The economists conducted their field experiment in the run-up to the 2022 presidential election in Colombia. Like the United States, Colombia is grappling with political polarization. Within a context of extreme tribalism, the authors suggest, truth becomes more disposable and the demand for misinformation rises. People become willing to believe and share anything in their quest for their political tribe to win.

To figure out effective ways to lower the demand for misinformation, the economists recruited over 2,000 Colombians to participate in an online experiment. These participants were randomly distributed into four different groups.

One group was shown a video demonstrating "how automatic thinking and misperceptions can affect our everyday lives." The video shows an interaction between two people from politically antagonistic social groups who, before interacting, express negative stereotypes about the other's group. The video shows a convincing journey of these two people overcoming their differences. Ultimately, they express regret over unthinkingly using stereotypes to dehumanize one another. The video ends by encouraging viewers to question their own biases by "slowing down" their thinking and thinking more critically.

Another group completed a "a personality test that shows them their cognitive traits and how this makes them prone to behavioral biases." The basic idea is they see their biases in action and become more self-aware and critical of them, thereby decreasing their demand for misinformation.

A third group both watched the video and took the personality test.

Finally, there was a control group, which neither watched the video nor took the personality test.

To gauge whether these nudges get participants to be more critical of misinformation, each group was shown a series of headlines, some completely fake and some real. Some of these headlines leaned left, others leaned right, and some were politically neutral. The participants were then asked to determine whether these headlines were fake. In addition, the participants were shown two untrue tweets, one political and one not. They were asked whether they were truthful and whether they would report either to social media moderators as misinformation.

What They Found

The economists find that the simple intervention of showing a short video of people from politically antagonistic backgrounds getting along inspires viewers to be more skeptical of and less susceptible to misinformation. They find that participants who watch the video are over 30 percent less likely to "consider fake news reliable." At the same time, the video did little to encourage viewers to report fake tweets as misinformation.

Meanwhile, the researchers find that the personality test, which forces participants to confront their own biases, has little or no effect on their propensity to believe or reject fake news. It turns out being called out on our lizard brain tribalism and other biases doesn't necessarily improve our thinking.

In a concerning twist, the economists found that participants who both took the test and watched the video became so skeptical that they were about 31 percent less likely to view true headlines as reliable. In other words, they became so distrustful that even the truth became suspect. As has become increasingly clear, this is a danger in the new world of deepfakes: not only do they make people believe untrue things, they also may make people so disoriented that they don't believe true things.

As for why the videos are successful in helping to fight misinformation, the researchers suggest that it's because they encourage people to stop dehumanizing their political opponents, think more critically, and be less willing to accept bogus narratives even when it bolsters their political beliefs or goals. Often — in a sort of kumbaya way — centrist political leaders encourage us to recognize our commonalities as fellow countrymen and work together across partisan lines. It turns out that may also help us sharpen our thinking skills and improve our ability to recognize and reject misinformation.

Critical Thinking In The Age Of AI

Of course, this study was conducted back in 2022. Back then, misinformation, for the most part, was pretty low-tech. Misinformation may now be getting turbocharged with the rapid proliferation and advancement of artificial intelligence.

List and his colleagues are far from the first scholars to suggest that helping us become more critical thinkers is an effective way to combat misinformation. University of Cambridge psychologist Sander van der Linden has done a lot of work in the realm of what's known as "psychological inoculation," basically getting people to recognize how and why we're susceptible to misinformation as a way to make us less likely to believe it when we encounter it. He's the author of a new book called Foolproof: Why Misinformation Infects Our Minds and How to Build Immunity . Drawing an analogy to how vaccinations work, Van der Linden advocates exposing people to misinformation and showing how it's false as a way to help them spot and to reject misinformation in the wild. He calls it "prebunking" (as in debunking something before it happens).

Of course, especially with the advent of AI deepfakes, misinformation cannot only be combated on the demand side. Social media platforms, AI companies, and the government will all likely have to play an important role. There's clearly a long way to go to overcoming this problem, but we have recently seen some progress. For example, OpenAI recently began "watermarking" AI-generated images that their software produces to help people spot pictures that aren't real. And the federal government recently encouraged four companies to create new technologies to help people distinguish between authentic human speech and AI deepfakes.

This new world where the truth is harder to believe may be pretty scary. But, as this new study suggests, nudges and incentives to get us to slow our thinking, think more critically, and be less tribal could be an important part of the solution.

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Thinking Of Buying An Electric Car? Here Are Five Critical Things To Consider

Range is one of them and so is your charging situation, but there are a few more things to consider before buying an ev..

W hile they still account for a slim percentage of the new-vehicle market in the U.S., sales of electric cars, led by the top-selling Tesla Model Y , are rising steadily. And with several new models constantly coming out and additional off-lease electric cars headed to used-vehicle lots, 2024 could be the year you take the proverbial plunge and buy an electric car. But shopping for a battery-powered vehicle is a somewhat different process than it is with a conventional gas-fueled model. Here are five prime variables to consider when buying either a new or used electric car.:

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Ensure That The Electric Car's Range Fits Your Needs

This is arguably an electric car shopper’s most important concern. Today’s electric cars cover a wide range in this regard, with driving capacities ranging from around 100 miles to more than 500 miles, with the sweet spot being the 250- to 300-mile range . Do you even need that much? 

It pays to pick a model with an operating range that exceeds your expected needs, and here’s why: Estimated ranges for electric cars are just that – estimated – and will vary based on several factors. For starters, you’ll drain the battery quicker while driving on the highway than around town. It takes more energy to propel an electric car at higher speeds.

Also, you’ll burn more kilowatts of power driving in extreme temperatures. Research conducted by AAA found that when the mercury dips to 20°F and the vehicle’s heater is in use, an average electric car’s range will be 41 percent shorter than when the mercury is at 70°F. When the thermometer hits 90°F, an electric car’s range drops by an average of 17 percent with the air conditioning running. And you’ll want to be prepared for the possibility that you may take on a new job that requires a longer commute at some point during your ownership period.

More EV 101 Info

• How Do You Maintain An Electric Car?
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Consider The Prices And Any Available Incentives And Tax Credits

Next to an electric car’s operating range, sticker price is the next most important consideration. The least expensive electric cars for 2024 are priced around the $30,000 range, though some luxury-minded models fetch well over $100,000.

You should buy or lease an electric car well within your financial reach. Always research prices for any vehicles you’re considering, including options, via an Internet source like Kelley Blue Book (kbb.com) or NADAGuides.com to see what you can afford. Also, consider the manufacturer’s mandatory delivery charge, registration costs, and state and local taxes that will be tacked onto the vehicle’s transaction price.

Remember that some electric cars sold in the U.S. are eligible for a one-time $7,500 federal tax rebate that effectively cuts the price by that amount.  On the other hand, those who lease an electric car can usually get the full benefit of this incentive immediately, as the credit is usually wrapped into the deal, with the leasing company being the owner of record. Several states offer their own tax breaks or cash rebates to help promote electric car sales.

If you’re on a tight budget, consider choosing a used model. With the exception of just a few models, which hold onto their resale values well, electric cars are incredibly affordable in the pre-owned market. They tend to have below-average resale values because of the aforementioned federal tax rebate, still limited demand, and other factors. The trade-off is that most three-year-old or older electric cars tend to deliver fewer miles on a charge than many of the latest models.

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Ensure That It's The Vehicle You Want

Aside from paying attention to an electric car’s price and range, you’ll want to choose a model you can live with as a daily driver. For example, a subcompact model’s interior may be too cramped for taller motorists. Some electric cars are inherently easy to enter and exit for some drivers than others. Back seat leg and headroom would be a concern if you regularly carry multiple passengers. If you have small children, you’ll want to determine how easy or difficult it might be to get them in and out of their car seats.

You’ll also want to make sure the cargo room in the trunk (if it’s a sedan) or behind the back seats (if it’s a hatchback or crossover sport-utility vehicle) can hold a sufficient number of grocery bags or cover your typical warehouse-store purchases. Also, check the cargo volume with the rear seatbacks folded flat, and note how easy or difficult it is to load and unload bulky items.

Basically, don't just go for any electric car because you want to drive electric. Pick an electric car that actually suits your wants and needs.

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Make Sure You Understand And Enjoy The Electric Car Driving Experience

No matter what type of vehicle you’re looking for, new or used, gas or electric, you’ll want to take a thorough test drive before signing a bill of sale. It’s essential to ensure the vehicle is acceptably comfortable, with all accessories easily operated and in good working order. Importantly, those who have never driven an electric car will notice that there’s a somewhat different driving experience involved.

Since there’s no engine or exhaust, an electric car is whisper quiet, which can be off-putting to some motorists. An electric car immediately delivers 100 percent of its available power, making higher-powered models feel twitchy until you learn how to modulate the accelerator.

Electric cars typically use one-speed transmissions, so there’s no sensation of gearshifts. Also, the vehicle’s regenerative braking function, which helps recover energy that would otherwise be lost through decelerating and braking, can be quite pronounced depending on the model. What amounts to a high level of engine braking may feel unnatural. Some electric cars allow the driver to choose more or less regenerative braking as desired and even enable so-called “one pedal” driving that will bring the car to a complete stop without using the brake pedal.

More Charging Info

• What Are The Different Types of EV Chargers?
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Consider Your Charging Options

Even if you have a standard power outlet in your garage, you’ll want to spend the money to have an electrician install a 240-volt line and perhaps a wall-mounted charger to take advantage of quicker “Level 2” charging times. While it can take 24 hours or more to fully charge a given battery-powered vehicle on house current (Level 1), most models can reach full capacity overnight via a Level 2 circuit. Check with your local utility company to see if they offer a discount for electricity during off-peak hours.

You’ll also want to note where public charging stations are located near where you live, work, and shop in case your car is running low on kilowatts while away from home. Look for “Level 3” DC Fast Charge installations that can bring an electric car’s battery pack up to 80 percent of its capacity in as little as 15 minutes, depending on the vehicle. However, public charging is neither as convenient nor economical as home charging.

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