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Teaching! (Planning ("Low-floor, high-ceiling tasks are also the most…
Teaching!
Planning
"Low-floor, high-ceiling tasks are also the most engaging and interesting math tasks." (p. 115)
Students need to be engaged in order to properly learn, and with so many students at different levels in a classroom it can be difficult to keep them all engaged at the same time. This is why high-floor and low-ceiling problems are so important, because all students no matter what level they are at can participate in the activity.
"Most of the unmotivated and bad behavior that happens in classrooms comes from students who do not believe that they can achieve." (p. 114)
Students NEED to know teachers believe in them, and teaching for a growth mindset is vital in a math classroom. Once students know you believe in them, they will try a lot harder and their bad behavior will often go away because they are distracted with learning.
"Multiple representations - students were frequently asked to represent their ideas in different ways, such as through words, graphs, tables, symbols, and diagrams." (p. 126)
Students need to know how to represent their answer and ideas in many different ways, as this shows they understand the connections between different math concepts. Therefore when planning, it is important to ensure problems can be solved and shown using a variety of different methods.
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"Inviting students to 'think about it' is appropriate for what is necessary, but not for what is arbitrary." (Hewitt p.5 )
If you ask students to think about what is arbitrary they will be confused, because this is something that they must simply be told as it cannot be discovered. Therefore, it is very important to know what is arbitrary and what is necessary when planning and teaching to prevent students from being confused and wasting valuable time.
"The arbitrary has to be memorized, but what is necessary is about educating their awareness" (Hewitt p.9 )
Arbitrary knowledge simply has to be told to students, but students can uncover necessary information for themselves by building off of previous knowledge if they are given the help and opportunity to do so.
"The 5 Cs of mathematical engagement" (curiosity, connection making, challenge, creativity, and collaboration) (p. 57)
When planning, it is important to consider the 5 Cs of mathematical engagement as by including these five things while lesson planning it will help teaching the material go smoothly because students will be interested and engaged in learning.
"When we pose problems for which students need to know a method before we introduce the method, we offer a great opportunity for learning and for using intuition." (p.81)
Helping students learn to use intuition helps them so much in the real world, because in the real world they will not be shown step by step how to do something. Instead, they will be expected to use prior knowledge to figure it out for themselves.
"When students are invited to ask a hard question, they often light up, totally engaged by the opportunity to use their own thinking and creativity." (p.85)
Certain students need more of a challenge than others, and an easy way to do this and ensure they understand the concepts they are learning is by getting them to come up with a similar (but harder) question once they finish all their work.
"My advice in bringing the real world into the classroom is to use real data and situations and to give a context only when it is helpful." (p. 200)
Students NEED to realize that math does exist in the real world, and is helpful when dealing with real world problems. Therefore, using real data and situations in the classroom shows students that math is important and will help them in the future.
Implementing
"In a growth mindset classroom the teacher is the one making these decisions in relation to individuals or groups of students, to challenge, support, and stretch them at exactly the right level." (p. 115)
It is VITAL that teachers know their students, and can see when they are struggling in a good way and just need to work through it, or when they are really struggling and need a little help to be able to continue working successfully. Teachers need to know what their students can handle, and what would discourage them.
"Giving rich tasks that the teachers described as group-worthy problems - problems that were difficult to solve alone and that required different members of a group to contribute." (p. 122)
When working, students need to have a challenge that requires them to depend on each other. If the problem is too easy, one member may dominate and complete the entire problem by themselves. Therefore, teachers need to ensure the problems presented keep students engaged, interested, and challenged and require them to work together.
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"Before any students work on math in a group with others, I ask them to discuss in small groups what they do and do not appreciate from others." (p. 173)
Group work is so important in mathematics because if students do not know how to collaborate and work together they will not be able to solve problems in the real world. However, before students can successfully work together they need to be taught how to collaborate respectfully with others.
"It is important not to provide too much help to students and take away from the cognitive demand of tasks." (p. 179)
Students need to struggle in order to learn, if they are always immediately given a hint of what step to do next in a problem they will never know how to work through a hard problem and try different methods until they find one that works and makes sense.
"We don't have students who formulate problems which is terrible" (Dan Meyers - Math Class Needs a Makeover)
Formulating problems is so important in math, because if you cannot formulate a problem you will never be able to complete math in the real world. Situations that need solving in real life are not formulated into a clean problem for you to solve using a particular formula. Therefore it is very important to allow students to formulate problems in school so they don't miss out on a huge aspect of true mathematics
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Assessing
"In complex instruction classrooms, teachers value, and assess students on, the many different dimensions of math." (p. 121)
Mathematics is not about getting to an answer using one way or method. There are SO many paths that students could take while exploring and solving a problem. Therefore, in my classroom I will allow my students to solve problems using any method that works for them and makes sense to them, because they should NOT be docked points for not solving the problem in a particular way.
"Adaptable, critical, and analytical thinking needed by students in the modern world." (p. 141)
Instead of using tests that require students to memorize certain methods in my classroom I will assess students using projects and group learning where they must use critical reasoning, analytical thinking, and adapt their solutions along the way as that is what workplaces desire from their workers, and I want to set all my students up to succeed in the real world, not just in the classroom.
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"Teachers who use A4L spend less time telling students their achievements and more time empowering students to take control of their learning pathways." (p. 149)
I realize that assessment FOR learning is key, as this helps students understand what they are learning, where they are succeeding, and what areas they can still work on without causing lots of stress by giving students a grade for their work. This is a wonderful assessment method to use to help students achieve at much higher levels.
"The most powerful learners are those who are reflective, who engage in meta cognition" (p. 150)
Reflecting on learning is so important, because during this time you make valuable connects and realize exactly what you just learned. Therefore, at the end of every class that I can I will put aside a certain amount of time for students to reflect on their learning in a reflective journal that they will use for the entire semester (or year)
- Allow students to resubmit any work or test for a higher grade
- Share grades with school administrators but not with students
- Use multidimensional grading
- Do not use a 100-point scale
- Do not include early assignments from math class in the end-of-class grade
- Do not include homework, if given, as any part of grading
(p. 167-168)
I want to ensure all my students can succeed to the best of their abilities, and give them every opportunity to increase their understanding. Therefore, incorporating these 6 methods for assessment are ideal to use in the classroom to help students have a growth mindset and achieve at a high level.
"Countries that gave more math homework had lower overall test scores than those that gave less math homework." (p. 107)
This shows me that homework is pointless, and I should not be giving it to my students as practicing the same thing over and over doesn't help them understand the mathematical process, it just helps them memorize steps to get to an answer.
For tests, write the final answers on the whiteboard so kids are not worried (class)
By doing this, students will not be stressed as much during tests in my classroom, and it shows them that I believe the process of finding the answer is way more important than the actual answer.
Blue - Quote and/or Idea from Class or Readings
Orange - Why I think the idea is important and/or how I will use it in my classroom