Presuming Competence

When interviewing students about their strategies and techniques when it appears they may be struggling, have an open mind. With that be asking questions and trying to find out what exactly is going on through the student’s work and explanations. When you have an open mind you are opening your understand of the student as a mathematical thinker. We should always presume competence. Presuming Competence, what is this exactly?

Presuming Competence means that we should always assume a person has the capacity to think, learn, and understand – even if you don’t see any tangible evidence that such is the case.

It is also very important to connect to the student’s thinking. Connecting with the student will push them toward new directions, and show them that it is okay to try new strategies, as well as how important it is for success to fail. No one ever gets anywhere without failing at least once. This type of understanding is almost like a work of art.

“and the art of working with students should always be centered on building off their own brilliance.”

Building Better Relationships

When teachers are told that students can achieve at a higher level, they believe in the students more. Teachers should always be looking for opportunities to provide feedback, both constructive and positive. When students hear “I believe in you” they achieve more in less time. There is so much unrecognized power in teacher-student relationships! All students deserve to hear positive beliefs from the teacher, especially those who could have underlying struggles in their lives.

“You can be the person who turns things around for students and liberates their learning path. It usually takes just one person – a person whom students will not forget”

There is so much value and importance in connections and relationships built with your students. No significant learning can happen without a significant relationship because kids don’t usually learn from people they do not like. Part of building these relationships is admitting to mistakes and apologizing for things done wrong, this will build trust within the relationship. Something else to keep in mind is to acknowledge all students’ victories, even if they are small. This will connect the student to you more as well as boost their self-esteem and confidence which will help them believe in themselves more. “Every child deserves a champion” in the words of Rita Pierson.

“A champion: Someone who will never give up on them, understands the power of connection and insists they become the best they can possibly be”

Growing your Brain

Did you know that when you believe in yourself, your brain operates differently than if you did not believe in your abilities. Everyone has a mindset, or a core belief about how they learn and what strategies work best for them.

Growth Mindset ~ You believe that smartness increases with hard work Fixed Mindset ~ You believe you can’t change your basic level of intelligence

When mindsets are changed and you start to believe that you can learn to high levels, you are actually changing your learning pathways and can achieve higher levels of learning. Growth mindset individuals have a greater awareness of errors than those with a fixed mindset. If you believe in yourself, your brain is more likely to spark and grow when mistakes are made. When a mistake is made, synapses fire in our brains.

A synapse is an electrical signal that moves between parts of the brain when learning occurs.

We do not need to be aware of the mistake that was made in order for our brains to spark. Making mistakes is not only a learning opportunity but can also lead to times where your brain can grow.

The first step to growing our brains and having sparks from synapses is to have an open mind and believe in the power of your own thinking!

Relational Understanding

This is understanding how and why the rules and procedures work. Students who are taught relational understanding usually have an easier time remembering certain procedures because they had a deeper meaning of why they work. These students will also retain all their learned knowledge longer and are also less likely to make common mistakes. Some students will be lacking motivation and determination and others may be too ready to accept your help because they are not used to thinking things through for themselves. Through their growth as learners students will make sense of math and will soon be less worrisome.

“Those with a relational understanding can learn new concepts easier, retain previous concepts, and are able to deviate from formulas/rules given different problems easier because of the connections they have made.”

When a student wants to make sense of learned concepts but are not given the time and conditions to experience math, they will come to believe that they are not good at math or they will say “they are not math people”. There are a few ways to try and remedy this; notice instrumental teaching, learn how to move from one to the other, and align assessment practices to meet with relational understanding. When you are unaware of your teaching style it was believe that you are doing a good job because your students are clearly learning something. By becoming more aware of your teaching style you can stop the behaviors that are like instrumental teaching and start transitioning to relational teaching.

The strands of Mathematical Proficiency

Conceptual understanding – comprehension of mathematical concepts, operations, and relations This strand is all about knowing more than isolated facts. Students should be linking lessons together and understanding the connections in math. They should also be able to correlate new ideas to previously learned lessons. Students also should be trying to represent mathematical situations in different ways and be forming deep, rich connections.

“Conceptual understanding frequently results in students having less to learn because they can see the deeper similarities between superficially unrelated situations. “

Procedural fluency – skill in carrying out procedures flexibly, accurately, efficiently, and appropriately. Students should be efficient and accurate when performing basic computations without needing aids or tables. Students need to see that procedures can be developed that will help solve more problems. Without sufficient procedural fluency students cannot deepen their understandings and can then start disconnecting from math. If students learn without understanding they could separate what happens in school from what is outside.

“In the domain of number, procedural fluency is especially needed to support conceptual understanding of place value and the meanings of rational numbers.

Strategic competence – ability to formulate, represent, and solve mathematical problems. Students are often presented with clearly specified problems to solve, outside of school they encounter situations in which part of the difficulty is to figure out exactly what the problem is. Students should also know a variety of solution strategies and should know what each one will be most useful in certain situations. It is also important for everyone to start by building a mental image of the essential components.

“Not only do students need to be able to build representations of individual situations, but they also need to see that some representations share common mathematical structures.”

Adaptive reasoning – capacity for logical thought, reflection, explanation, and justification. This is the glue that holds everything together and that guides learners. Students who are disagreeing about an answer need to not questions anyone or anything except for their own reasoning, and whether it is valid or not. Some people say that kids do not start deep reasoning until 12 years old, but four and five-years old can explain their reasoning of how they solved the problem. A skill in this strand would to be able to justify your work. Some teachers truly believe that there is only one way to do something and being able to explain how you reached an answer can help everyone learn something new.

“Students need to be able to justify and explain ideas in order to make their reasoning clear, hone their reasoning skills, and improve their conceptual understanding.”

Productive disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with belief in diligence and one’s own efficacy. This skill can be built through frequent opportunities to make sense of mathematics, to see the benefits of determination and to experience the joy of math making sense. When students see themselves as capable of learning and using mathematics, they can begin to develop more layers of their other strand skills. These students will usually be confident in their knowledge and ability.

“The more mathematical concepts they understand, the more sensible mathematics becomes.”

https://www.nap.edu/read/9822/chapter/6#131