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-These are all levels within the structure process that also devise mathematical processes.
-However BOTH t he Knowledge and Structure Processes are DIFFERENT they each support one another in the quest of learning.
-For example: Students need to have the skill of looking at two points on the coordinate plane and identify it's slope. In order to accomplish this skill they must also be able to pull from their mind the fact that to find slope we have a very specific formula to help us find the slope of a line.

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The last of the 6 key processes is investigation which means we allow students to explore certain situations they have never encountered before.
-This is a great opportunity for students to learn how to use resources and observe how creative they can be when they encounter challenging math problems.
-Do they try to solve it alone? Use collaboration as a resource? Maybe they get creative and recall a real world event that helps them? Allowing them to have the time to investigate on their own can lead to many discoveries they may have not been aware of until that precise moment!

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-It is crucial that math gets represented visually as much as possible for students, regardless of the type of learners you may have.
-Technology can be your best friend in helping achieve this goal as well as other visual tools like, guided notes, graphic organizers, manipulatives, actual 3D figures, Nearpod, even small clips on YouTube.

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-Crucial for students to develop the ability to make connections between stuff they have learned in the past with the new concepts they are currently learning.
-As an educator it is our job to make sure they also connect what we teach them to the real world.
-I gave my students time to process during lessons once I realized that if I moved too quickly they missed on making connections because they might feel rushed.

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-This is a process that I personally focus heavily on this past school year because I feel in math it is very easy to forget to stop and have dialogue.
-I used guided notes, stations, and other activities in my classroom to provide students the opportunity in class to discuss and collaborate amongst themselves about terms, problems and what they believed they meant, etc.
-This was what I believe our chapter reading this week referred to as Social Constructivism because we constantly made discoveries via interactions student-student and teacher-student.

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-One of the major things that I gathered from this key process is the ability to make mistakes and the power this holds within learning.
-Students need to view mistakes as an important weapon to gain more knowledge, especially in math.
-What I used this last school year was partner checks. One student would do the problem and then their partner with a color pencil would need to check their work and correct any mistakes. This then allowed the student who might have made an error the ability to see their error and learn from that mistake.
-Lastly, reasoning and proof is can be connected with generalizations.

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-When students are expected to problem solve the goal is that they can understand a problem , know the plan to do so, carry that plan out and hopefully execute to come to the correct solution.
-Problem solving is a very important skill that students must develop to better understand the math for example guess and check or drawing things out are some I have used in the past to help my students.

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-DIFFERENT from one another.
-Formulas basically represent cold hard FACTS within the mathematical world.
-Theorems are proven STATEMENTS which lead to conceptual understanding in math.

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-As an educator myself, I now realize I tend to use generalizations quite often in the classroom.
-Generalizations help develop critical thinking and develop inquiry skills within the learning environment as well.
-For example, I will present a statement to the class when doing a lesson that allows my students to make connections about concepts within the problem we may be covering.

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Making generalizations is fundamental to mathematics. Developing the skill of making generalizations and making it part of the students' mental disposition or habits of mind in learning and dealing with mathematics is one of the important goals of mathematics education. Making generalizations is a skill, vital in the functioning of society. It is one of the reasons why mathematics is in the curriculum. Learning mathematics (if taught properly) is the best context for developing the skill of making generalizations.

What is generalization?

There are three meanings attached to generalization from the literature. The first is as a synonym for abstraction. That is, the process of generalization is the process of "finding and singling out [of properties] in a whole class of similar objects. In this sense it is a synonym for abstraction (click here to read my post about abstraction). The second meaning includes extension (empirical or mathematical) of existing concept or a mathematical invention. Perhaps the most famous example of the latter is the invention of non-euclidean geometry. The third meaning defines generalization in terms of its product. If the product of abstraction is a concept, the product of generalization is a statement relating the concepts, that is, a theorem.

For further discussion on these meanings, read Michael Mitchelmore paper The role of abstraction and generalization in the development of mathematical knowledge. For discussion about the importance of generalization and some example of giving emphasis to it in teaching algebra, the book Approaches to Algebra - Perspectives for Research and Teaching is highly recommended. There is a chapter about making generalizations and with sample tasks that help promote this attitude.

Sample lessons

Mathematical investigations and open-ended problem solving tasks are ways of engaging students in making generalizations. The following posts describes lessons of this type:

1. Sorting number expressions

2. Lesson study: Teaching subtraction of integers

3. Math investigation lesson: polygons and algebraic expressions

4. Polygons and teaching operations on algebraic expressions

Of course it is not just the type of tasks or the design of the lesson but also the classroom environment that will help promote making generalization and make it part of classroom culture. Students will need a classroom environment that allows them time for exploration and reinvention. They will need an environment where a questioning attitude is promoted: "Does that always work?" ,"How do I know it works"? They will need an environment that accords respect for their ideas, simple or differing they may be.

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Making generalizations in mathematics - Mathematics for Teaching

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Making generalizations is fundamental to mathematics. Developing this skill and making it part of the students' mental disposition or habits of mind...

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-The lowest level
-For example students all recall the Pythagorean theorem but they don't theoretically understand it.
-Most students of my Geometry students could remember the formula we needed when I would present them a right triangle but didn't know why we needed to use it or how it works.

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Understanding Primary Math by S. Sundaram

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From Understanding Primary Math by S. Sundaram

Factual part & conceptual aspects of math knowledge are two levels in which math can be learnt. Both constitute knowledge in math that need to be "remembered & recalled".

Factual knowledge can be considered the "final" form in which a deep concept is communicated to another person who has already understood the underlying concepts.

To take a simple example, conceptual knowledge is about "understanding" why a particular number is even. Factual knowledge is remembering a rule by which we can decide if a number is even. The important point is that one may now the rule but not why that rule works.

Conceptual knowledge is the deeper aspect. We will deal with this kind of knowledge in the next chapter.

Here are some statements in math that can be called factual knowledge.

1. 2, 3, 5, 7 are prime numbers less than 10.

a. The learner may not know what a prime number is Or

b. Whether 97 is a prime number

2. 8 X 7 = 56

a. The learner may not know the meaning of this statement

b. The learner may be unable to provide any real-life event which can justify it.

3. Odd + Odd = Even

a. The learner may know the meaning of the terms even & odd but may not know how to justify the above statement.

4. A Square is a quadrilateral with all its sides and angles equal.

a. The learner may know this definition without understanding the relation between a square and another quadrilateral like rectangle

5. A Right Angle equals 90 degrees

a. The learner may not understand the above statement at all.

b. The learner may not know the relation between a right angle & a complete angle

6. While evaluating an expression involving arithmetic operations, the multiplication & division operations have to be done before the addition & subtraction operations.

a. The learner may not know why such a rule is needed.

b. The learner may not know the meaning of a convention

Factual knowledge consists of (a) statements, (b) definitions & (c) conventions.

Statement (4) is a definition & statement (6) is a convention. The others are statements.

These statements are typical of the answers to some of the questions asked in math tests. They can be written as answers to questions.

What is important is that they can be answered, without understanding what they mean. We will explore the meaning of "understanding" when we deal with conceptual knowledge.

The statements themselves may or may not be mathematically correct. Their "validity" cannot be verified by a person who does not have conceptual understanding.

Typically, the Indian school curriculum emphasizes mostly on factual knowledge in contrast to conceptual knowledge.

Since factual knowledge consists of definitions, statements & conventions, which have not been really understood, the standard pedagogy is to memorize them.

Most school examinations contain questions which can be answered "correctly" without even understanding the meaning of either the question or the answer.

This is why you can have a student who scores very high marks in the school-leaving examinations but finds difficult to understand post-school math which is built on conceptual understanding expected to be developed in school.

Factual knowledge is the easiest aspect of math to be learnt.

This is one of the reasons, the curriculum and teachers focus on this aspect of knowledge in any subject.

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From Understanding Primary Math by S. Sundaram Factual part & conceptual aspects of math knowledge are two levels in which math can be learnt. Both constitute knowledge in math that need to be "remembered & recalled". Factual knowledge can be considered the "final" form in which a deep concept is communicated to another person who has already understood the underlying concepts.