Meso concepts are those more specialized areas in those disciplines. They tend to be a bit more niche than the main disciplines. Such as trigonometry & discrete math.

Micro concepts tend to be the specific concepts of each subject. For Algebra some examples are system of equations, graphing quadratics, or finding the value of certain variables.

Theorems are in statement form & work as proofs for math concepts. There are explanations for why things work & it's heavily tied to conceptually understanding the "why." These theorems work as principles to our math concepts.

Strategies often involve use of many different math skills, these skills are used in ways that will help develop their understanding of a problem.

Algorithms also involve use of certain skills, but in a more structured manner. Properly using an algorithm often leads to a specific outcome.

We've mentioned them a lot, but what are skills? As we see skills are engrained into our algorithms/strategies. These skills are small actions that allow these algorithms/strategies to work properly. Such as knowing your order of operations, knowing how to plot a line for a linear equation, etc.

This example explains how the proper use of the order of operations can help someone solving a many different problems of varying complexity. These concepts can also be used to help convey how it may be used in real life applications.

This example highlights the usage that logarithms can have instead of simply stating it as an operation. The key points in the generalization are that logs are an additive process as opposed to the inverses (exponential functions) multiplicative process.

• Can refer to beyond one specific concept in math (they're versatile).

• Are truths supported by factual examples.

• May not always be true, as they're some exceptions.

Note: Video discusses both generalizations below (yellow boxes)

• Define a particular concept in math.

• Work as proofs for math concepts, meaning they should be true in every case.

• Always have qualifiers in their statements.

Level 2 generalizations are our teaching targets for our the instruction we give. These go into the how & whys for the concepts students learn. This level of generalizations should be where the students gain the most understanding (not entirely application) of the topics. They get an idea of the importance of the topic, at level 3 they get to a point of application.

Level 3 generalizations while not mandatory can help build students depth of what they learned. Showing them a bigger scope of the usage of a concept, highlighting its importance. How may it be used in the real world? After giving the level 3 generalization it's a great idea to follow with real life examples students may see in their day to day lives.

Some ways to do this are letting them put their math thoughts into writing, graphic organizers, concept maps, & much more.

Example:

How many sides (outward facing) are added each time you add a cube next to another (in a horizontal straight line)?

Four sides are added each time you add a cube next to another.

Why are four sides added each time (outwardly) you add a cube next to another (in a horizontal straight line)?

The reason four sides are added each time is because when cubes are put together, the "connected sides" between the cubes disappear. So the only sides that remain are the top, bottom, front, & back for each cube (besides the end cubes which have five sides).

Note: The intro was pretty uncanny for me haha.

Below are 3 of the visible thinking routines I enjoyed & would use in my class.

Example of a Do & Don't for solving linear equations. DO: Balance the equation before you solve for a variable (especially when the equation is multi-step). DON'T: Subtract 3 when you should divide by 3 to isolate X from 3X.

Roles: What the student will do in the project

Audience: Who will read/is affected by the project

Format: What medium students will use to represent their project

Topic: What is being learned & applied

Strong: Usage of words that will hook students .

These projects can have a huge range. Like finding the volume of school buildings, or using linear equations to represent how far railroad ties are from each other on a track.

Zero, One, Two, or Three: This strategy is a quick, easy, & subtle way to check your student's understanding of what was learned. Students write a number from 0-3 on paper. Ranging from completely lost (0) to completely understood (4). You can use this to check in with your students mid lesson & see where to go from there. It can also be used on student's homework, which I personally would love to implement. Even though many students may complete the homework, there are many different ways they could have gotten to that point. Whether it was with a lot of struggle or relative ease. This would help me know who needs further assistance.

Self management relies on students being able to pace themselves in their own learning & being motivated to do so. This one is a bit tricker but I think it starts in building students confidence in math. It's hard to get started when you don't know where you should be going. This is why scaffolding is important. You can provide easier problems that go up in difficulty or complexity. By starting this way you can help students they're prepared for what's coming next. You can also start with a hard question, but I must emphasize you have to be VERY careful if you do so. If done wrong you can damage student's self confidence. Show the questions as a way of representing an end point. Have them think about it & bring in thoughts, don't focus on their ability to solve the hard question. Assess where students are at & build from there. Reassure your students that you will work gradually to that end point & you'll be there for support every step of the way. When they get to the point where they're ready & do solve it there confidence will increase, knowing that they were able to overcome the once seen impossible.

Research skills in math are greatly helped by student's own curiosity. Provide questions that get them hooked, that leave them asking for more knowledge (this doesn't mean to not provide an answer). Admittedly this is much easier said then done & even I'm having difficulty imagining how I can get students to that point.