I hate to say this: I used the lesson plan and resources today, Sept. 30th 2014. I felt the lesson went really well, the children were v enthusiastic, and worked with real concentration. Unfortunately, the lesson was being observed by the Head and Deputy. The preliminary feedback (more tomorrow) is that the lesson isnt "age appropriate" for Years 5 and 6. Will keep updated on further feedback.........
Thanks for the resources brillaint lesson! Some help needed though- I am definately over thinking this but what is the final formula for the Nth term? Thanks
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For everyone whose relationship with mathematics is distant or broken, Jo Boaler , a professor at Stanford Graduate School of Education (GSE), has ideas for repairing it. She particularly wants young people to feel comfortable with numbers from the start – to approach the subject with playfulness and curiosity, not anxiety or dread.
“Most people have only ever experienced what I call narrow mathematics – a set of procedures they need to follow, at speed,” Boaler says. “Mathematics should be flexible, conceptual, a place where we play with ideas and make connections. If we open it up and invite more creativity, more diverse thinking, we can completely transform the experience.”
Boaler, the Nomellini and Olivier Professor of Education at the GSE, is the co-founder and faculty director of Youcubed , a Stanford research center that provides resources for math learning that has reached more than 230 million students in over 140 countries. In 2013 Boaler, a former high school math teacher, produced How to Learn Math , the first massive open online course (MOOC) on mathematics education. She leads workshops and leadership summits for teachers and administrators, and her online courses have been taken by over a million users.
In her new book, Math-ish: Finding Creativity, Diversity, and Meaning in Mathematics , Boaler argues for a broad, inclusive approach to math education, offering strategies and activities for learners at any age. We spoke with her about why creativity is an important part of mathematics, the impact of representing numbers visually and physically, and how what she calls “ishing” a math problem can help students make better sense of the answer.
What do you mean by “math-ish” thinking?
It’s a way of thinking about numbers in the real world, which are usually imprecise estimates. If someone asks how old you are, how warm it is outside, how long it takes to drive to the airport – these are generally answered with what I call “ish” numbers, and that’s very different from the way we use and learn numbers in school.
In the book I share an example of a multiple-choice question from a nationwide exam where students are asked to estimate the sum of two fractions: 12/13 + 7/8. They’re given four choices for the closest answer: 1, 2, 19, or 21. Each of the fractions in the question is very close to 1, so the answer would be 2 – but the most common answer 13-year-olds gave was 19. The second most common was 21.
I’m not surprised, because when students learn fractions, they often don’t learn to think conceptually or to consider the relationship between the numerator or denominator. They learn rules about creating common denominators and adding or subtracting the numerators, without making sense of the fraction as a whole. But stepping back and judging whether a calculation is reasonable might be the most valuable mathematical skill a person can develop.
But don’t you also risk sending the message that mathematical precision isn’t important?
I’m not saying precision isn’t important. What I’m suggesting is that we ask students to estimate before they calculate, so when they come up with a precise answer, they’ll have a real sense for whether it makes sense. This also helps students learn how to move between big-picture and focused thinking, which are two different but equally important modes of reasoning.
Some people ask me, “Isn’t ‘ishing’ just estimating?” It is, but when we ask students to estimate, they often groan, thinking it’s yet another mathematical method. But when we ask them to “ish” a number, they're more willing to offer their thinking.
Ishing helps students develop a sense for numbers and shapes. It can help soften the sharp edges in mathematics, making it easier for kids to jump in and engage. It can buffer students against the dangers of perfectionism, which we know can be a damaging mindset. I think we all need a little more ish in our lives.
You also argue that mathematics should be taught in more visual ways. What do you mean by that?
For most people, mathematics is an almost entirely symbolic, numerical experience. Any visuals are usually sterile images in a textbook, showing bisecting angles, or circles divided into slices. But the way we function in life is by developing models of things in our minds. Take a stapler: Knowing what it looks like, what it feels and sounds like, how to interact with it, how it changes things – all of that contributes to our understanding of how it works.
There’s an activity we do with middle-school students where we show them an image of a 4 x 4 x 4 cm cube made up of smaller 1 cm cubes, like a Rubik’s Cube. The larger cube is dipped into a can of blue paint, and we ask the students, if they could take apart the little cubes, how many sides would be painted blue? Sometimes we give the students sugar cubes and have them physically build a larger 4 x 4 x 4 cube. This is an activity that leads into algebraic thinking.
Some years back we were interviewing students a year after they’d done that activity in our summer camp and asked what had stayed with them. One student said, “I’m in geometry class now, and I still remember that sugar cube, what it looked like and felt like.” His class had been asked to estimate the volume of their shoes, and he said he’d imagined his shoes filled with 1 cm sugar cubes in order to solve that question. He had built a mental model of a cube.
When we learn about cubes, most of us don’t get to see and manipulate them. When we learn about square roots, we don’t take squares and look at their diagonals. We just manipulate numbers.
I wonder if people consider the physical representations more appropriate for younger kids.
That’s the thing – elementary school teachers are amazing at giving kids those experiences, but it dies out in middle school, and by high school it’s all symbolic. There’s a myth that there’s a hierarchy of sophistication where you start out with visual and physical representations and then build up to the symbolic. But so much of high-level mathematical work now is visual. Here in Silicon Valley, if you look at Tesla engineers, they're drawing, they're sketching, they're building models, and nobody says that's elementary mathematics.
There’s an example in the book where you’ve asked students how they would calculate 38 x 5 in their heads, and they come up with several different ways of arriving at the same answer. The creativity is fascinating, but wouldn’t it be easier to teach students one standard method?
A depiction of various ways to calculate 38 x 5, numerically and visually. | Courtesy Jo Boaler
That narrow, rigid version of mathematics where there’s only one right approach is what most students experience, and it’s a big part of why people have such math trauma. It keeps them from realizing the full range and power of mathematics. When you only have students blindly memorizing math facts, they’re not developing number sense. They don’t learn how to use numbers flexibly in different situations. It also makes students who think differently believe there’s something wrong with them.
When we open mathematics to acknowledge the different ways a concept or problem can be viewed, we also open the subject to many more students. Mathematical diversity, to me, is a concept that includes both the value of diversity in people and the diverse ways we can see and learn mathematics. When we bring those forms of diversity together, it’s powerful. If we want to value different ways of thinking and problem-solving in the world, we need to embrace mathematical diversity.
Maisuna Kundariati , Rahel Natalia Saragih Munthe , Maria Rosalia Ijung Anggur , Herawati Susilo , Balqis Balqis , Frida Kunti Setiowati; Promoting biology students’ scientific reasoning in plant physiology lessons using problem-based learning (PBL). AIP Conf. Proc. 24 May 2024; 3106 (1): 030031. https://doi.org/10.1063/5.0214972
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Scientific reasoning skills now days become discussed among scholars because its importance in every daily life. As a part of 21 st -century skills, it is necessary to promote it in science courses, especially in biology lessons. The purpose of this research is to enhance students’ scientific reasoning skills. Students who participate in this research were enrolled for the plant physiology course. The research was conducted during September-December 2022 at Universitas Negeri Malang. The learning process was carried out through Problem-based Learning. Classroom learning is done by implementing lesson study to improve each learning process. The research instruments were chapter design. Lesson design, lesson plan, and essay test of scientific reasoning skills refer to the rubric of scientific reasoning by the AACU (2010). The scientific reasoning achievement of students in the Plant Physiology course has increased from the initial, middle, and final tests. The average achievement of scientific reasoning on the initial, middle, and final tests, respectively, is 54.17, 63.13, and 72.56. The n-gain score of the scientific reasoning skills of students is 0.65 in the medium category. Meanwhile, the achievement of students’ scientific reasoning on the Topic or Argument Selection indicator is 0.58 in the medium category; Existing Knowledge, Research, and/or Views of 0.75 in the high category; Analysis of 0.64 in the medium category; and Conclusions, Limitations, and Implications of 0.57 in the medium category.
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Achieving Excellence in Teaching and Learning
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10 May 2024
Outstanding High Potential and Gifted Education (HPGE) Mathematics Day organised by 'Inquisitive Minds’ at Plumpton High School proved to be a highly effective way for representing schools to provide high quality and fun maths activities for their students.
Plumpton High School's outstanding performance was notable, as they secured first place in the competition. Plumpton High School was also declared the 'Inquisitive Minds Talented Maths Champions' of the Greater Western Sydney Region, solidifying their ability in mathematical problem-solving and teamwork.
Students from Plumpton High School achieved individual success, with Year 9 students Year 9 students Ysabela Marasigan and Lanvy Nguyen earning 1st place and Jericho Villareal and Jordan Villareal placing third. Year 8 students Taha Ali and Zakariya Al-Shible also achieved winning scores while collaborating with students from another school. In the Engineering Competition, Plumpton High School secured second place, with the group led by Dev Patel constructing a bridge with a remarkable span of 57.5m.
The event exposed HPGE maths students from Year 8 and Year 9 to four hours of problem-solving activities, including an interactive lesson and engineering competition. It enabled advanced problem-solving strategies and critical thinking skills to be developed and practiced. With participation from four schools (Riverstone High School, St Johns Park High, Jamieson High School and Plumpton High School), the day facilitated collaborative learning and practical application of mathematical concepts, demonstrating the value of hands-on, interactive experiences in promoting mathematical proficiency.
A huge thank you to Ms Prasad, Ms Zhang and Ms Deni-Savio for organising the successful event and other schools for their participation in healthy competition that has built complex mathematical reasoning and problem solving.
Tim Lloyd, Principal
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Here are five strategies to help students check their solutions. 1. Use the Inverse Operation. For simpler problems, a quick and easy problem solving strategy is to use the inverse operation. For example, if the operation to solve a word problem is 56 ÷ 8 = 7 students can check the answer is correct by multiplying 8 × 7.
not a book of sample math lesson plans. There are oodles of math lesson plans available on the Web and from other sources. By and large these lesson plans have three weaknesses: 1. They are not personalized to the individual strengths and weaknesses of the teacher, the teacher's students, and their culture. 2.
Then, I provided them with the "keys to success.". Step 1 - Understand the Problem. To help students understand the problem, I provided them with sample problems, and together we did five important things: read the problem carefully. restated the problem in our own words. crossed out unimportant information.
Teaching about problem solving begins with suggested strategies to solve a problem. For example, "draw a picture," "make a table," etc. You may see posters in teachers' classrooms of the "Problem Solving Method" such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no ...
Step 1: Understanding the problem. We are given in the problem that there are 25 chickens and cows. All together there are 76 feet. Chickens have 2 feet and cows have 4 feet. We are trying to determine how many cows and how many chickens Mr. Jones has on his farm. Step 2: Devise a plan.
Problem-Based Tasks Require Studentsto Apply Their Knowledge in New Contexts. Problem-based tasks are math lessons built around a single, compelling problem. The problems are truly "problematic" for students — that is, they do not offer an immediate solution. The problems provide an opportunity for students to build conceptual understanding.
engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies. ... skillful in using procedures flexibly as they solve contextual and mathematical problems. Support productive struggle in learning mathematics. Effective teaching of ...
Problem Solving. This feature is somewhat larger than our usual features, but that is because it is packed with resources to help you develop a problem-solving approach to the teaching and learning of mathematics. Read Lynne's article which discusses the place of problem solving in the new curriculum and sets the scene.
Implement Tasks That Promote Reasoning and Problem Solving Most of NCTM's attention to task selection and implementation focuses on the cognitive demand of mathematical tasks (Smith & Stein, 1998). Tasks with low cognitive demand are those that simply require memorization or ask students to perform a procedure without connecting that ...
Introduction. You can use either Explicit Instruction or Self-Regulated Strategy Development when you intervene to support your student's problem solving skills. The following lesson plan targets a specific problem-solving skill using explicit instruction. As you read this plan, consider: How does this plan support objective mastery? Problem ...
Math Lesson Plans for Teachers. Filter. Sort by: Most-Popular Relevance; Most Popular; Most Recent; Most Popular. x Mathematics. x Lesson Plans (85) results found ... Using a Formula is a problem-solving strategy that students can use to find answers to math problems involving geometry,… Subjects: Mathematics. Formulas and Functions. Download ...
Math Reasoning Lesson Plans. Math Sleuths Students use their problem-solving skills to complete an online Math Hunt. Their share their answers and justify their thought processes with the class. Math in Everyday Life Students work with a partner to come up a with word problems, involving time, money, and simple fractions. They then share their ...
Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted ...
Through problem-solving and mathematical modeling, teachers can encourage deeper thinking. ... Mathematical reasoning skills are a core part of critical thinking. Through problem-solving and mathematical modeling, teachers can encourage deeper thinking. ... Lesson Plan Outline. An example that might be appropriate for fifth grade is something ...
Introduction. (10 minutes) Bring students together in a circle, either seated or standing. Bring blocks with you to the circle. Show the student the blocks and ask them to watch you build a tall castle. After you build it, bring out two figurines that you would like to play with in the castle. Say out loud, "Hmm....there seems to be a problem.
Instructional programs from prekindergarten through grade 12 should enable each and every student to—. Recognize reasoning and proof as fundamental aspects of mathematics. Make and investigate mathematical conjectures. Develop and evaluate mathematical arguments and proofs. Select and use various types of reasoning and methods of proof.
Mastering Subtraction Word Problems - Lesson Plan. In this interactive math lesson, students will develop problem-solving skills as they learn strategies to solve subtraction word problems. Through engaging activities and real-world scenarios, students will gain a deeper understanding of subtraction and its application in everyday life. 1.
The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to: (B) use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution and evaluating the problem-solving process and the ...
Mathematical Reasoning 2014 GED® Assessment Targets Quantitative Problem Solving Standards and Content Indicators Develop and understanding of ratio concepts and use ratio reasoning to solve problems. (CCR.MA.ABE.8.3.1) Explain ratio concepts and use ratio reasoning to solve problems (CCR.MA.ABE.8.4.1) Analyze proportional relationships and ...
Mathematical Reasoning & Problem-Solving Critical Thinking and Logic in Mathematics 4:27 Logical Fallacy | Overview, Types & Examples 4:47
Maths Problem Solving Lesson Plan Year 5 and 6. Problem Solving, patterns and sequences, algebra. This lesson achieved outstanding by Ofsted today. SMartboard file is based on a resource I found on Primary Resource called 'Magic Carpet'. Please e-mail me with any questions: [email protected].
Free Google Slides theme, PowerPoint template, and Canva presentation template. Let's make math learning more fun, especially at early levels of education. This new template has some cute illustrations and lots of elements related to math, including backgrounds that look like blackboards. This is a great choice for teachers who want to turn ...
The case for 'math-ish' thinking. In a new book, Jo Boaler argues for a more flexible, creative approach to math. "Stepping back and judging whether a calculation is reasonable might be the ...
Despite the growing emphasis on integrating collaborative problem-solving (CPS) into science, technology, engineering, and mathematics (STEM) education, a comprehensive understanding of the critical factors that affect the effectiveness of this educational approach remains a challenge.
The learning process was carried out through Problem-based Learning. Classroom learning is done by implementing lesson study to improve each learning process. The research instruments were chapter design. Lesson design, lesson plan, and essay test of scientific reasoning skills refer to the rubric of scientific reasoning by the AACU (2010).
The event exposed HPGE maths students from Year 8 and Year 9 to four hours of problem-solving activities, including an interactive lesson and engineering competition. It enabled advanced problem-solving strategies and critical thinking skills to be developed and practiced.