"To conceptualize a number as being composed of two or more parts is the most important relationship that can be developed about numbers (p. 152). Your assessment should be looking for the following progression of understanding: can students accurately divide a number into two addends (first step); can students accurately divide a number into many addend pairs; can students develop a complete set of addend pairs; do students include 0 and the number as a possible pair.
Formative assessment (observations) can be done when students are working on any of the activities provided in the book. Early proficiency, however, is most easily assessed with Activity 8.18. I like the suggestion of asking students to write the addends down on a record sheet - the act of writing engages a different part of the brain and helps with retention. Alternatively, enough manipulatives could be provided that students can simply make an inventory of all addend pairs they have made for a given number.
This visual log of what students have accomplished allows you, the teacher, to engage in a discussion with students about their pairs. They may be missing some possible pairs (the target number and 0 is often missing, for example). You can ask them if their pairs could be ordered or arrange in a way that creates a pattern.
Activity 8.19 may be a bit more advanced, but it would allow for spot checking student understanding.
A more formal assessment could be done using proportionate part-part-whole cards or a worksheet. A card is divided into three parts: the top half represents the whole; the bottom half is then divided proportionally into two parts. Any two of these three cells can be filled in with appropriate numbers and students can be asked to complete each card.
Your writing about formative assessment I too think is critical. Also the ability to keep track of the assessment is difficult for me even though I value the idea. Entrance tasks help me because I collect them and am able to then take a closer look at them. The number sense of some of my current students in sixth grade is an indication that they could have benefited from so many of the ideas presented in this chapter. Picking up number patterns, thinking and talking about numbers, subitize, to place value with the numbers and have the students understand the importance that is related to the size of the number or doubling of the number, or the lessening or lose of value with numbers. Currently trying to review and reteach sixth grade students to develop number sense while trying to also move them forward is a daunting task. I also am questioning what should be done when students fail to reach important grade standards and the amount that needs to be taught is pulling me to teach the next goal in the curriculum.
I have had similar experiences with family background influencing student opinion and effort in class. One student, just moments after watching a Jo Boeler video, said that she wasn't a math person. As an intern I was fairly clear with her that I didn't want to hear her say that again - everyone can be a math person. I then took it upon myself to find things she was good at, in math and other classes, and build rapport with the student. I never hear her say those words again, but her smile let me know she was thinking them. Eventually, we both found math skills she was good at, and the teacher and I focused on those successes to build confidence.
I agree that having students write down the addends is great in engaging different parts of their brain and thinking. I have worked with younger students who can easily identify that they need to add 3+2 to get 5 but need to recount and double check before writing the problem out. I think this is a natural progression of the student understanding the part-part-whole relationship.
I think that mernickg's (sorry, I do not know who you are yet) question is really a valuable one that I've spent a lot of time considering. When I used to substitute teach, I would often end up spending lots of time helping the 3-5 students in the classroom catch up, while everyone else was ready to move on. I struggle with what to do in this kind of situation. I don't know yet the pressure that real classroom teachers have to teach the curriculum and do so in a timely fashion; I only know that this problem causes me to lose the rest of the students' attention and focus, and allows for lots of time to misbehave. Perhaps classroom teachers could enlist the help of special educators or aides, though I am not sure how practical this is. It shows me how important it is that students really understand mathematical foundations at an early age, because without that, they have no knowledge to build upon and they will always struggle. ~Meghan Ehle
This was a really fun read for me as I currently work with a kindergartner and I was able to make a lot of connections!
A teacher can assess part-whole number relationships by first checking to see if a student is able to understand and complete the basic ingredients of part-part-whole activities. This includes the student being able to compose two or more addends to reach a number, or decompose a number into two or more parts. Additionally, students can be asked to state the problem that they have created by composing or decomposing a number and then writing that problem down.
The activities in the book provide great stepping stones to reach a complete understanding of part-whole relationships. While these activities are being completed, a teacher may record observations and discuss with the student how they reached an answer and how they knew which steps to take. From this, a teacher can see if a student understands and is comfortable with the flexibility of numbers.
I agree with Tiffany that teacher observations and discussions will provide valuable insight into a students understanding of the material being assessed.
(Disclaimer: I did not read anyone else's comments first, so I am sorry if this has already been said; I didn't want to be influenced before I wrote this out). A teacher can assess part-whole number relationships by playing a game, such as the "covered parts" activity. The teacher begins with an certain amount of counters (representing whole numbers) that she/he knows the student has knowledge about. She hides some under a cup or paper, asking what is missing and how does the student know? This goes on, with higher numbers each time, until the student begins to struggle. That is a good stopping point, so the student does not get frustrated. The teacher can record the highest number that the student knows automatically to assess the students knowledge of single digit whole numbers and the process by which they subtract to get the answer and add to check it. ~Meghan Ehle
I think that by engaging a student in any of the activities listed a teacher can assess their understanding of the Part-Part-Whole relationship. I think the activity "Missing-Part Cards" is an excellent example that requires a higher understanding the parts that make up a number.
A teacher could use many methods for assessing understanding of part-part-whole relationships of numbers. One way I particularly like is observing students as they create patterns with different colored beads or beans. This could be done by having white beans and red beans one each child's desk. The teacher holds up a number under 12 and asks children to create a part-part-whole relationship with the different beads. When student's have completed it on their desks, invite one student per question up to the overhead projector to share "their" answer with their classmates. Another way, is to ask kids to write a part-part- whole equation on a slip of paper with their name on it. Have them drop it in a hat on the way out the door.
This is where you will share your response to the question and/or your response to your classmates' responses.
ReplyDeleteWe're looking forward to seeing your responses.
ReplyDeleteType your thoughts here.
ReplyDelete"To conceptualize a number as being composed of two or more parts is the most important relationship that can be developed about numbers (p. 152). Your assessment should be looking for the following progression of understanding: can students accurately divide a number into two addends (first step); can students accurately divide a number into many addend pairs; can students develop a complete set of addend pairs; do students include 0 and the number as a possible pair.
ReplyDeleteFormative assessment (observations) can be done when students are working on any of the activities provided in the book. Early proficiency, however, is most easily assessed with Activity 8.18. I like the suggestion of asking students to write the addends down on a record sheet - the act of writing engages a different part of the brain and helps with retention. Alternatively, enough manipulatives could be provided that students can simply make an inventory of all addend pairs they have made for a given number.
This visual log of what students have accomplished allows you, the teacher, to engage in a discussion with students about their pairs. They may be missing some possible pairs (the target number and 0 is often missing, for example). You can ask them if their pairs could be ordered or arrange in a way that creates a pattern.
Activity 8.19 may be a bit more advanced, but it would allow for spot checking student understanding.
A more formal assessment could be done using proportionate part-part-whole cards or a worksheet. A card is divided into three parts: the top half represents the whole; the bottom half is then divided proportionally into two parts. Any two of these three cells can be filled in with appropriate numbers and students can be asked to complete each card.
Your writing about formative assessment I too think is critical. Also the ability to keep track of the assessment is difficult for me even though I value the idea. Entrance tasks help me because I collect them and am able to then take a closer look at them. The number sense of some of my current students in sixth grade is an indication that they could have benefited from so many of the ideas presented in this chapter. Picking up number patterns, thinking and talking about numbers, subitize, to place value with the numbers and have the students understand the importance that is related to the size of the number or doubling of the number, or the lessening or lose of value with numbers. Currently trying to review and reteach sixth grade students to develop number sense while trying to also move them forward is a daunting task. I also am questioning what should be done when students fail to reach important grade standards and the amount that needs to be taught is pulling me to teach the next goal in the curriculum.
DeleteI have had similar experiences with family background influencing student opinion and effort in class. One student, just moments after watching a Jo Boeler video, said that she wasn't a math person. As an intern I was fairly clear with her that I didn't want to hear her say that again - everyone can be a math person. I then took it upon myself to find things she was good at, in math and other classes, and build rapport with the student. I never hear her say those words again, but her smile let me know she was thinking them. Eventually, we both found math skills she was good at, and the teacher and I focused on those successes to build confidence.
DeleteI agree that having students write down the addends is great in engaging different parts of their brain and thinking. I have worked with younger students who can easily identify that they need to add 3+2 to get 5 but need to recount and double check before writing the problem out. I think this is a natural progression of the student understanding the part-part-whole relationship.
DeleteI think that mernickg's (sorry, I do not know who you are yet) question is really a valuable one that I've spent a lot of time considering. When I used to substitute teach, I would often end up spending lots of time helping the 3-5 students in the classroom catch up, while everyone else was ready to move on. I struggle with what to do in this kind of situation. I don't know yet the pressure that real classroom teachers have to teach the curriculum and do so in a timely fashion; I only know that this problem causes me to lose the rest of the students' attention and focus, and allows for lots of time to misbehave.
DeletePerhaps classroom teachers could enlist the help of special educators or aides, though I am not sure how practical this is.
It shows me how important it is that students really understand mathematical foundations at an early age, because without that, they have no knowledge to build upon and they will always struggle.
~Meghan Ehle
This was a really fun read for me as I currently work with a kindergartner and I was able to make a lot of connections!
ReplyDeleteA teacher can assess part-whole number relationships by first checking to see if a student is able to understand and complete the basic ingredients of part-part-whole activities. This includes the student being able to compose two or more addends to reach a number, or decompose a number into two or more parts. Additionally, students can be asked to state the problem that they have created by composing or decomposing a number and then writing that problem down.
The activities in the book provide great stepping stones to reach a complete understanding of part-whole relationships. While these activities are being completed, a teacher may record observations and discuss with the student how they reached an answer and how they knew which steps to take. From this, a teacher can see if a student understands and is comfortable with the flexibility of numbers.
I agree with Tiffany that teacher observations and discussions will provide valuable insight into a students understanding of the material being assessed.
Delete(Disclaimer: I did not read anyone else's comments first, so I am sorry if this has already been said; I didn't want to be influenced before I wrote this out).
ReplyDeleteA teacher can assess part-whole number relationships by playing a game, such as the "covered parts" activity. The teacher begins with an certain amount of counters (representing whole numbers) that she/he knows the student has knowledge about. She hides some under a cup or paper, asking what is missing and how does the student know? This goes on, with higher numbers each time, until the student begins to struggle. That is a good stopping point, so the student does not get frustrated. The teacher can record the highest number that the student knows automatically to assess the students knowledge of single digit whole numbers and the process by which they subtract to get the answer and add to check it.
~Meghan Ehle
I think that by engaging a student in any of the activities listed a teacher can assess their understanding of the Part-Part-Whole relationship. I think the activity "Missing-Part Cards" is an excellent example that requires a higher understanding the parts that make up a number.
ReplyDeleteA teacher could use many methods for assessing understanding of part-part-whole relationships of numbers. One way I particularly like is observing students as they create patterns with different colored beads or beans. This could be done by having white beans and red beans one each child's desk. The teacher holds up a number under 12 and asks children to create a part-part-whole relationship with the different beads. When student's have completed it on their desks, invite one student per question up to the overhead projector to share "their" answer with their classmates. Another way, is to ask kids to write a part-part- whole equation on a slip of paper with their name on it. Have them drop it in a hat on the way out the door.
ReplyDelete