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Pg. 83
"We should choose meaningful
representations in which the objects and actions available to the student parallel
the mathematical objects (ideas) and actions (processes or algorithms) we wish
the students to learn" (Sarama & Clements, pg. 83)

"High-achieving children build images that have a spectrum of quality
and a more conceptual and relational core. They are able to link different
experiences and abstract similarities" (Sarama & Clements, pg. 83).

Not just having manipulatives, but as educators helping facilitate the use of them and allowing children to explore them. The objects should not just be a mechanical things (for children to memorize them and their use) but help them draw parallels and real world connections.

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"What ultimately makes mathematical ideas Integrated-Concrete is not their physical
characteristics, but how “meaningful”—connected to other ideas and situations—
they are" (pg 76)

- It is important to understand that the manipulatives are there as supplement/support to the actual mathematical concept. Without understanding the concept that is trying to be taught, the manipulative becomes impractical.

"An idea is not either concrete or not concrete. Rather, depending on how you
think about it, on what kind of relationship you have with it..."  (pg 76)

- This quote shows the importance of providing prior knowledge to ensure the relationship with the manipulative is effect

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"We conclude that manipulatives are meaningful for learning only with respect to learners’ activities and thinking and that both physical and virtual manipulatives can be useful. "

It is significant because involving manipulatives in the teaching is to help students to better understand the mathematics concept, but if students just follow the instructions of how to use the manipulatives instead of truly understanding the concept itself, it will not be effective.

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Most stu- dents can use physical manipulatives to perform motions such as slides, flips and turns; however, they make intuitive movements and corrections without being aware of these geometric motions. Even young children can move puzzle pieces into place without any attention to the geometric motions that can describe these physical movements. Using virtual tools to manipulate shapes can bring those geometric motions to an explicit level of awareness (Clements and Sarama 2007a).

we found this quote significant because it highlights the effectiveness of manipulating geometric shapes using virtual tools. Every motion the student implements upon the shape is more intentional because it is less intuitive.

Implication for teaching: using virtual tools for geometry learning is incredibly effective, allowing the students to manipulate on devices like chrome books will benefit their awareness of geometric motions.

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"manipulatives do not guarantee meaningful learning and may not need to be physical"

"This sensory nature is assumed to make manipulative "real," connected with one's experience in the physical world, and therefore foundational to learning"

Manipulatives don't always facilitate learning, so it should be done in a way where  students can make connections between manipulative and their own experiences.
Implications: We need to make sure students can make the connections, not us.

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"Most teachers recognize that if young children have not explored a manipulative on their own (say, toy dinosaurs), getting them to address the teachers agenda (say, counting) can be at best inefficient, and at worst, a serious struggle".

"These few manipulatives should be used for multiple tasks, so children do not view them as objects to play with but tools for thinking" (Sowell 1989).

Implication for Teaching:
Recognizing that students necessarily do not need a lot of manipulatives in the classroom and that a few are sufficient. A few manipulatives would be helpful for students to learn and not see this as a form of play or unstructured learning.

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"Thus, it was not simple “concrete”experience that helped but rather attention to quantities. Concrete objects may play an important role but they need to be used carefully to create a strong understanding and justification for each step of a procedure" - pg. 3

Further in the article, the author talks about teaching the idea of a math concept/topic instead and being careful of the number of manipulatives you use and how you use them. Therefore, manipulatives should be used with instructions and modelling to ensure students are able to connect the idea of a math concept to the manipulative. Without this scaffolding further confusion and misunderstanding can occur.

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"Although manipulatives may often play an important role in learning, their
physicality does not carry the meaning of the mathematical idea."
We chose this quote because it places importance on manipulatives but also implies that math must be understood properly before physical elements are introduced. Students must think about the role of manipulatives in their work and comprehend why using them helps them apply significance in their learning.

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"One possibility is that instruction does not adequately promote connection
between children’s representations based on manipulatives and those based on
paper and pencil" (e.g., Carnine et al. 1997; Sherman and Bisanz 2009).

This is significant because it is saying that using physical and concrete manipulatives are important, but if the students don't understand the value or representation of those manipulatives, then they do not benefit from using them.

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“Surprisingly, manipulatives do not have to be physical objects” (Sarama and Clements, 2016, p.77)

It’s not the fact that we can touch it, but rather can students make connections with the manipulative

IMPLICATION FOR TEACHING: seeing the value of the manipulative, encouraging students making meaningful connections rather than seeing manipulative as a symbol

Example: seeing a 100 block as pure a block vs students making connection that 100 block actually represents a value.

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"The key is that they are successful because they can model the situation" This quotation discusses how students are more successful when completing tasks when using manipulatives versus without. This is significant because students are able to see the work in front of them instead of having to navigate mental math while still in the process of learning whatever it is, they are learning (in this case counting).

"One sequence of studies showed that 3-year-olds who used more “interesting” manipulatives (fruit instead of plain blocks) were more likely to accurately identify numbers in a recall task" students are able to process manipulatives better when using things they can relate to everyday life. We think this relates to most concepts in math as students are more likely to be engaged when they are learning with items related to their lives.

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1. "This is consistent with research showing that most children do not solve larger-number problems without the support of concrete objects until 5.5 years of age."

2. "Such material often facilitates children’s devel- opment of mathematical operations by serving as material support for children’s action schemes."

The significants is that students are required to have an understanding of concrete objects. This is required knowledge that is needed before applying to more difficult number problems. It is import that we as educators are aware of this need for tangible materials that may assist in students success in mathematics.

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1) abstraction is not to be avoided at any age - math is about embracing abstraction and generalization to categorize concepts for children to understand better.
2) often true that younger children possess the relevant knowledge but cannot effectively create a mental representation of the necessary info - this is significant because good manipulative can effectively assist them in understanding these mathematical concepts.

Implications: abstract ideas are very difficult to explain and for children to understand - teaching using concrete examples benefits students learning. All children are capable of understanding concrete concepts - not just older children with previous math knowledge - assistance from manipulatives can help them understand greater concepts

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"Manipulatives, drawings, and other representations should as much as possible, be used instructionally in ways consistent with the mental actions on objects that students are to develop."

"relationship of manipulatives to the concepts they are to represent is not transparent to children"

The manipulatives and objects students use can be very effective tools for learning, if they are first understanding the concepts. If the concept is not understood it can hinder students learning.

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Physical and Virtual Manipulatives - What is _concerete__