Magical+Hopes

By Deborah Ball

How is it that Sean thought 6 could be both odd and even?

Using the definitions given to Sean and his classmates about odd and even numbers, "a number is even if you can split it in half without having to use halves, a number is odd if you have to split one in half";it appears that Sean doesn't understand **__one half.__** He seems to know that when one counts by 2's (2, 4, 6), these are the even numbers. By helping Sean differentiate between one half, meaning 2 equal parts of a whole, and grouping the objects in groups of 2; he may be able to conceptualize the difference between the two concepts. Once he understands this concept, he can understand why 6 can not be both odd and even. He has made 3 groups of two objects and not 2 equal groups of the whole number 6. -EI

I would explain to Sean that odd and even refers to the digit. My students had drew a number card from a pile, counted out the amount, then if one was left after partners divided then they determined that the digit was odd. If their was no counters left. the digit was considered even. However, some students still did not understand the concept. It took several additional weeks of practice for them to understand the concept. --FS Sean heard groups and split it into different groups. He did not understand that he had to be able to divide the whole in 1/2. He instead tried to divide the whole into even groups. JJ,CS

We can see this at our school. Our students do not have a basic understanding of numbers. If they are using counters or other manipulatives they can see the general idea but without manipulatives they seem to lose the understanding of the idea. DJ, JV, DH

Odd and even are really difficult concepts for young kids. I am constantly on the lookout for new ways to teach this concept. But is odd/even really something that is valuable for us as adults? We inherently know if a number is breakable into two equal or unequal parts. I rarely, if ever, consider the odd-ness or even-ness of a number. AP

Why were manipulatives not the answer for Sean?

He was able to use the manipulatives to establish his point of view but he lacked the necessary understanding of the concept being taught. He needs more experiences with varied materials, small group activities and discussions, and teacher modeling before using the manipulatives. -EI He used the manipulatives to prove his point. He did not understand the concept of even and odd. The manipulatives were really of no help to him. The manipulatives may have been helpful had Sean clearly understood the concept of even and odd numbers. JJ, CS He still would have held fast to his idea, even with the use of the manipulatives. -DD
 * Sean problems needed to understand the concept of digits and needs more time to comprehend the concept. FS**

He was convinced that his inference was correct and could have easily made manipulatives work to his advantage. AP

What assumption do we make as teachers about the connections that students make with the mathematics?

As we observe students working with manipulatives, we "assume" that they have acquired the mathematical understandings we were hoping they had learned..This is often not true. They can be able to manipulate the objects but their lack of understand the intended concepts become evident when asked to explain or show the concept using other mathematical strategies in the absence of manipulatives. -EI

Sometimes manipulatives help and sometimes they only confuse students, especially when we over use them and then take them away and expcet them to cover over the understanding to paper and pencil. DJ, JV, DH We assume that they are thinking how we think, and we forget about all of the experience/education we have...assuming that if we "see" it, so do our students!-DD


 * We assume that most children understand when they speak the same language as we do. We also forget that some students have not exercise that part of their brain power to grasp as quickly as we think they should. FS**

Teachers rarely are able to check in with each and every student after a teaching moment or lesson. We do need to remember that there are misconceptions at so many levels. Pre- and post-assessments could help. But I am NOT talking about formal assessments. We have to remember that assessing on the fly is part of the process. AP

We assume they make the same assumptions that we make or that we may want them to make. JJ, CS  We assume if they can do it one way they can do it anyway. And that is not always the case. - CMM

We assume that students know what to do with the manipulatives and how to associate concepts with the manipulatives. Unfortunately many teachers do not know the connect that needs to be established with the manipulative and can't explain the connection between the concept and the manipulative to the students. TW
 * How can we help students see the relationship between 4/8 and 4/4 and 3/3 and 5/5?**

As AP mentioned, I use items my children relate to and enjoy (things to eat). In addition, we draw pictures, sort items in groups using paper plates to show the relationship of quantity and proportional reasoning as they compare the number of objects each "student" has to show the relationship between 4/8 and 4/4. This activity is often done in cooperative learning groups where I can "listen in" to their "math talk" as they work. Teacher modeling using students and distributing the items helps them create mental models as they compare/contrast the number of items each received, helping them see the relationship between 4/8 and 4/4. Writing and drawing is important to show each student's understanding of the concept. -EI

We cannot rely on manipulatives alone...we must have "math talk" and use the appropriate math vocabulary, along with using models.-DD
 * I agree that we must model, talk, and let students practice with manipulatives, let them explain to a peers and write about what they understand about the concept and the process of problem solving. FS**

By using models and discussing them. Having the students explain what they are doing/seeing. JJ, CS

We spend a lot of our math time using math talk as well as journaling. DJ, JV, DH We often use items that kids can relate to, such as candy, pizza or other items that can be divided into appealing groups. And I agree with the others, it is important to connect math talk, writing and manipulatives. Group work can also help because group members bring many ways of looking at a problem to the table. AP Model, model, model. Demonstrate. Let them explain their own thinking. Guide understanding. - CMM

What do we need to keep in mind as we use manipulatives as a mathematical tool in developing conceptual understanding?

As we saw with Sean, the use of manipulatives alone can lead to misconceptions. We must create a learning environment where children can work cooperatively, sharing ideas and experimenting with various materials, as they solve mathematical problems. Teachers must help students make the connections from the concrete objects to the abstract mathematical concepts. -EI

We need to keep in mind that manipulatives serve the purpose of moving student thinking from concrete to abstact. Manipulatives are a vital part of development that is necessary for most learners. Manipulatives help gain interest and motivate. As humans we have a need to touch. We must know our students and realize when to move from manipulative to drawings to the abstract. Each child is unique and time varies for conceptual understanding. Students talking about amongst themselves has proven to be very powerful. -FS

Manipulatives are not the only thing to use. They are only a tool and need to be used as such. JJ,CS Manipulatives serve as a tool for developing conceptual understanding and learning in young learners. Manipulatives are a vital part in transferring learning to abstract thinking. Manipulatives serve all learners: kinesithic, visual, auditory. I don't believe manipulatives teach the students, however they are vital in student success. -aw We cannot rely on manipulatives alone. They are only part of our toolbox. AP We need to understand that using manipulatives alone will not lead to solving a problem. We must know how to use the manipulatives correctly in order to find a solution to a problem. However, using manipulatives help some students understand. -CMM