Chapter Reviews
Chapter 6
Math topics that perform well to performance assessment tasks
I think a math topic that will lend itself to do well in a performance assessment task is fractions. Fractions will work well because we use fractions a lot throughout life, so coming up with real world problems for context can really help students. Students can also practice their generalizations, test what they know adn test their skills.
Other Graphic Organizers that I use
Frayer Model
I do like to use the Frayer Model to ask questions. A lot of time, students like to expect a yes or no question, but that isn't good enough. I like students to think about their own examples to come up with, instead of agreeing or disagreeing.
How would I use a visible thinking routine
I would use a visible thinking routine before and during a lesson. I like the think, pair, share routine because it gets students engaged at he beginning of class, and during class.
At the beginning of class, I can use this to help refresh students' minds about the day before. I can say "I want you to think about setting up a division problem." They can think about it, discuss it with a partner to see if they forgot about anything, then share with the class.
During class, I can use it to have students think about the next step to do to solve an equation problem. I can say "I want you to think about the next steps I have to do in order to solve for X."
Advantages of visible thinking routines
Think, Pair, Share!
This allows students to take a moment to pause and think, then discuss with a partner, then share with the class. This is useful because I can see what students are struggling, and what students are understanding the concepts.
Can be used as bell work at the beginning of class
Allows students to express and communicate their personal thoughts and ideas.
Assessment Strategies
Agree, Disagree, Depends
When - I can do this during class, during a lesson. I like this a lot because it makes sure that every student is participating. I can also do this when I want to do a quick check for understanding.
Why - It is important to do this because students have a choice that they have to make. It also allows me to check what percentage of the class thinks what, and I can have them explain why.
How - I can do this by having students do a thumbs up, down, or to the side. I can also make dedicated cards for each student so that they can raise and no one else can see their answer. This will help with following what friends say/do, and help reduce embarrassment.
Assessments With Conceptual Depth
When - I can implement this every time I do an assessment. I don't want to know whether my students have memorized the formula or rules, I want to know if my students DO understand the concept.
Why - A lot of the time, students can do well in their class until it is time to take a test. They know the rules and formulas to everything, however they do not know how to "use" it properly. This is why word problems and having rigorous assessments are crucial.
How - I can do this with assessments to check their understanding. A really quick way to do this is to do a short 5 question quiz that tests their understanding/knowledge.
Open Ended Questions
When - I will use this when I want to know if my students have mastered the current concept.
Why - it is important for students to master the concept so that they can further improve their knowledge, and move onto more "challenging" concepts/ideas.
How - I can implement this by having them think about a question, then write down their answer while also providing evidence for their answer.
Chapter 3
Generalizations vs Principles
Principles
Principles are kind of like the foundation, or the bedding of a truth/true statement. Principles are more of that than generalizations, as principles include generalizations, but not vice versa.
Usually the rules we learn in math are principles.
Generalizations are usually two or more concepts that are in a way, related to each other.
Generalizations also have factual evidence to support their claim/statement, but do not always include every single situation.
Why do we wish our students to understand generalizations and principles in mathematics?
We want our students to understand generalizations and principles in mathematics for many reasons.
We want our students to understand the concepts that we are teaching them.
We want our students to know when a theory applies to a situation.
We want our students to know when a principle is "law" or when a generalization "can/may apply."
Level 1, 2, and 3 Generalizations
Level 3 Generalizations
Level 3 generalizations dig even deeper than knowing how all the parts work.
After knowing how the concept or theory works, the students start to learn what exactly is this "concept/theory" used for. Students may ask "why is this useful to know" or "know that I know this, WHAT can I use this for."
I think about level 3 as not only knowing, but also applying.
Level 2 Generalizations
Level 2 generalizations provide more "clear statements of understanding."
Level 2 generalizations are the part in which students spend their time learning and making new connections about the concept.
Level 1 Generalizations
Level 1 is very weak, and we often just want to skip over them.
Instead of saying how one thing may relate to another, we ask questions.
Questions we ask may include how or why. This gives our learners a chance to think deeper about a concept, and see how all the gears work together.
Opportunities that I Provide
Most commonly, I provide students the chance (or I cold call on students) to share the steps/rules of a concept (Level 2). Then, later on once I feel as if my students have a very good grasp on the topic, I give them the chance to share a real life example (Level 3).
I also give my students some time to talk to their partner where they can discuss about what we are learning. An example is "Turn to your partner, and discuss the rules/steps of what we are learning about," or, "Turn to your partner and discuss why/when can we use this in real life."
Chapter 8
What is the role of a teacher?
I think the role of a teacher is to both transmit information & knowledge, and to inspire, guide, and facilitate the social process of learning.
A teacher has a lot of roles to fill. Students look towards us for guidance and inspiration. At the same time, it is our job to share with them information and knowledge (it may be content information and knowledge, or even 'wisdom')
What does ideal math teaching look like?
I am providing clean and clear modeling for my students.
I am providing differentiation for the different levels that my students may be at
I have an engaging lesson to ensure that students are focusing and engaged
I am constantly asking questions to provide deeper thought.
What does ideal math learning look like?
Students make mistakes and learn from them. They are interested in why they did not get the answer correct the first time, or even second time.
Students asks questions to further clarify any confusing information
Students try their best during assessments
Students are engaged in the classroom
Using the rubric as a concept-based teacher.
Lesson Closing
I want to assess what my students know. To do this, I can have them complete an exit ticket that will give me a quick glance of what my students know.
It is important to have some time to reflect as a class. Going over what we learned, and having students ask some final questions is important.
Targets During the Lesson
During the lesson, I want students to be constantly thinking/listening. If they are not listening to what I have to say, they should be thinking about what I have said/presented.
I want to make sure that I have multiple ways to teach the lesson. Some students may find it a lot easier to follow along and learn if I have multiple ways of presenting the content to them.
I want to make sure that I model correctly. I cannot expect students to do work by themselves and succeed without first showing them how to do it.
Lesson Opening
I think the most important part of lesson opening is to get their attention. Getting their attention is a part of having an engaging lesson opening. If it is not engaging, then my students will not be interested.
How will I transfer a growth mindset in my classroom?
I can emphasize not giving up, and to always try your hardest. I can show this by having students learn from their mistakes. When students learn from their mistakes, they learn that learning is a constant thing that happens, and people aren't born "smart." If they put in the effort, I can even have them make up some homework, or more commonly a test/quiz to reach a better score.
Chapter 4
Encouraging Collaboration When Unit Planning
First, I can make sure that everyone on my team has something to contribute towards. If people have something that they can contribute towards, then that means they have something to work.
Second, I can have a unit web in which my peers will be able to share ideas. Sharing ideas is important because it allows a discussion to happen, which in turn helps the whole "project."
How Factual, Conceptual, and Debatable Questions Support Student Learning
These questions help support student learning by connecting "the factual support with the conceptual understandings."
Each type of question helps students to think about a certain generalization in a different way, thus, helping them to further understand each generalization.
Main Components of a unit web
Knowledge Generalizations
"Concepts in Mathematical Processes"
This includes the generalizations (what we learned in chapter 2) that are specific to the unit at hand.
Important Topics of Said Unit
How Unit Planning Helps Empower Teachers
Teachers get to take control of the learning. They decide what learning goes into the lesson planning, and decide (or try to guide) what are the learning outcomes.
Unit Planning is basically like a plan. With a plan, you get to decide what happens, and at what time it happens.
Chapter 2
Macro, Meso, and Micro
Macro Concepts
Macro concepts are concepts that contain a very large information of specific type of math.
Meso Concepts
Meso concepts are broken down from Macro concepts, into more specific concepts. One example is geometry being broken down into trigonometry.
Micro Concepts
Meso Concepts are then broken down further into micro concepts. Micro concepts are like skills.
Some examples include knowing ratios and angles. These are like skills because we use these concepts to use and understand geometry.
Formulae Vs. Generalizations
Generalizations
Generalizations are understandings of a concept that usually allows people to make connections with two or more things.
Formulae
Using mathematical symbols to provide a factual statement.
Mathematical Processes, Skills, and Algorithms
Algorithms
Algorithms consist of a set of skills. When following rules, these skills help solve mathematical problems.
Skills
Skills are kind of like the foundation of processes and strategies. Without any skills, one wouldn't know how to solve a mathematical problem.
Processes
Processes include strategies, algorithms, and skills.
Using prior knowledge to solve unusual problems, or thinking outside the box is an example of a process.
Structure Process
Creating Representations
Having different representations to connect/relate with math is very important.
Having visuals is another way to explain something.
I can teach exponents, but my students wouldn't really know how much faster exponents grow. If I have a graph, then they can actively see the rapid growth for themselves.
Making Connections
All math is connected in one way or another. Multiplying is another way of adding, and dividing is another way of subtracting, with all of these operations using the same numbers.
Finding out how other operations or processes work is very important so that we can make these connections. For example, I cannot multiply if I cannot add. When I am first introduced to multiplying, it is very beneficial to know what In am essentially adding/skip counting.
Communicating
Communicating includes talking and explaining, and also listening discussing.
If I only know what I am thinking, then there is no other way that I can learn. However if I talk to and listen to people, then I can see where they are coming from, and perhaps find a new perspective.
Reasoning and Proof
In order to know if you are right, you need to be able to provide explanations for your answer.
Reasoning is the first step. You must be able to reason in order to think about how to solve a problem.
Proof comes second when you need to actively work towards the problem. If you are correct, you need to have proof to why you are correct.
Problem Solving
The necessary part of Mathematics.
The learner uses skills to try to find solutions in situations where things may be difficult.
The Structure of Knowledge and Process
https://www.youtube.com/watch?v=CpJNOxPp5gY
Structure of Knowledge
The Factual Level
The lowest level in the Structure of Knowledge.
Facts are seen as more specific examples, and are often seen as a place, idea, time, people, or even a situation.
