Week 8: Discussion Question (Option 1)

12 Jan 2018 4:52 PM | Anonymous member (Administrator)

How does Emily's story (194-200) make you think about the role of connections in students' proficiency, or lack thereof?

Comments

  • 13 Jan 2018 12:21 PM | Anonymous member
    Unfortunately, I fell behind and I am trying to get back on track....
    With middle school students (special ed setting) I see the opposite of Emily. They know the facts, the fractions but they do not know how to visualize it and they cannot connect to a story. I spend a lot of time asking students to draw a visual to talk about where they would see this. I also spend time asking students to read the problem to me ie. 10 – x = 2, students want to jump ahead and give an answer; I call this ‘football math’, the quarter back recites a series of numbers which only mean something to the other players. Students need the connections in order to become proficient.
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  • 13 Jan 2018 4:46 PM | Anonymous member
    Connections are so important but equally difficult to determine. If a student gets the problem correct there is no guarantee they made a connection. Likewise, getting a problem wrong does not indicate complete misunderstanding. The other day I had students trying to write an equation to show how a pattern was growing. One of the things they mentioned when I asked what they noticed about the pattern was that "two was added every time." When the equation was created, I asked why we mentioned that 2 was added every time but in our equation the 2 was multiplying the variable. They were sure that the equation must be wrong. When I asked them what it meant to multiply, they could not explain it to me in terms of addition. Students are not making connections on their own unless we have conversation and look at problems from different view points. As teachers we need to be willing to use our limited class time to develop these connections.
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    • 14 Jan 2018 9:52 AM | Deleted user
      Ellen, your first two sentences really struck a chord in me - learning/assessing is very quirky and definitely not black and white.
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    • 15 Jan 2018 8:51 AM | Anonymous member
      I agree with you that students often do not make connections on their own. Sometimes the connections seem so incredibly obvious to us, but it is not obvious to them. I think your point about making time for conversations is imperative to helping students make those connections. I feel a big part of our job is to orchestrate the questions that lead the conversations that help students make those leaps and discover those connections for themselves...much more powerful than teachers making the connections for them.
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  • 13 Jan 2018 7:56 PM | Anonymous member
    I agree that making connections is important in helping students deepen understanding and become proficient in math. In the younger grades, I have experienced children not understanding the mathematical symbols and content specific vocabularyas much as they should. Beginning to regularly use content vocabulary, while modeling, is crucial. Although preparation for testing is not, or should not be, the goal, we are required to test students. I have found that young students often lack knowledge of essential vocabulary as well as a solid grasp of what symbols mean and how to represent their meaning in different ways.
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  • 14 Jan 2018 9:54 AM | Deleted user
    Emily’s story makes me think about the balance between inquiry and explicit instruction about connections. I have always struggled with this balance. Emily has “intuitive” number sense but was lacking some basic understanding. Her story gave me hope that once the gaps are correctly identified we can help “tangled” learners can untangle their thinking! The power of formative assessment! Imagine if Emily were only assessed on the fact sheet. Or, in this world of standardized testing, would her strengths be identified and built upon by a test result – I recognize this is a rhetorical question!!!
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  • 14 Jan 2018 11:56 AM | Anonymous member
    I find the opposite with most of my students. They can work with numbers, but if a story is attached, they have a harder time understanding what to do. I try to do a lot of work with numberless word problems to get the students to think about what is going on in the story. We discuss what is happening (joining, separating, finding differences, etc). Then, the students are given numbers (or can choose numbers) to work with. Once they figure out the solution, they write the equation(s) used to arrive at the solution. In this way, they connect the story to the mathematical representations.
    This is definitely one of the more difficult topics, I find, because I think so much emphasis has been/is on procedures. More time needs to be spent building connections and understanding. Our curriculum is still pretty wide so there is not enough time to go deep. There needs to be time to build those connections.
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  • 14 Jan 2018 4:05 PM | Deleted user
    I work with advanced students at the K-8 level. Unlike Emily, they are very good with "naked" numbers, and remember facts fairly easily. They might prefer to omit the stories as they have a more difficult time with word problems, where they need to decide from a context what operations to use with the numbers in the problem. I try to give them lots and lots of time to explore word problems so that they can gain both familiarity and confidence. Likewise, I try to give them time for inquiry based learning, where they 'discover' number or shape relationships. Students of all ages and all abilities need to have opportunities to explore new mathematics and to make their own connections between what they already know and what they are learning.
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  • 14 Jan 2018 6:33 PM | Anonymous member
    I teach 5th grade and I generally see the opposite of what Emily was showing. My students tend to know a lot of facts, but they don't know why the facts work. They have just memorized them. They recognize the math symbols and know what to do with them. It is hard to get 5th graders to write or work with word problems as they don't really want to spend time thinking about the math. they just want to get it done. I liked the idea of having the students represent their math thinking with pictures and then present them. I would like to try this with my class as I wonder how many of them really understand the math they are doing and how many of them are just going through the motions of the algorithms they have memorized.
    I still see many students in my class counting on fingers. I wonder how many of them have ever taken the time to really know what the number 7 or any number is. If they did I think grouping and regrouping would be so much easier for them and adding and subtracting would make so much more sense to them.

    I think this role of connections is extremely important and we as teachers need to spend more time on it. The struggle is we feel pressure to get through curriculum in a year, so teacher push on without taking the time to be sure students are truly proficient and we also have many behavior problems in the classroom that did not exist in the past and often it is hard to work one on one with the students who need that extra time as the rest of the class is ready to move on and is not good about working independently.
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    • 15 Jan 2018 12:22 AM | Anonymous member
      The choices we make to delve deeper/longer or to move on to cover more before the year ends should be considered seriously. We should feel pressure to do the best for the current students, not the day of the week, timing of a test or textbook assigned.
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  • 15 Jan 2018 12:19 AM | Anonymous member
    Whoa! That was scary! I would have moved her on to something else, then she would have suffered later because of "gaps". Multiple ways need to show that connections are made. That was quite a story to learn from.
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  • 15 Jan 2018 11:08 AM | Anonymous member
    I read this chapter before checking out the prompts and when I look back I see I wrote in the margin “some of my students” when the author talks about Emily. To be honest - about ¼ of my math class is like this! I think the students that are proficient are the ones making connections. And to be honest, this ¼ of the class is extremely reluctant to come in for one-on-one assistance/instruction leaving just class time to help them with their connections. By the time I have a conversation with one student about their work - another student needs assistance/clarification and then to have another conversation with another about their lack of connection…. there are just some students that truly benefit from from one-on-one clarification but at the same time other students have questions too. This ¼ of my class has a long history of not meeting proficiency - I need help in learning how to help them at the same time I am helping the students that are proficient move along.
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  • 15 Jan 2018 12:00 PM | Anonymous member
    When I was reading Emily's story it really got me thinking about a student in my classroom and ther lack of connection which is effecting their ability to show proficiency. One of the next steps for Emily was to work with objects and pictures to connect numbers to quantities. The student I have in mind struggled at the beginning of the year counting and adding within 10. I did alot of number work that had the student practice counting objects in different ways. Over the course of this year, the student has improved with their number sense and can fluently solve problems that only involve numbers as well as basic word problems. Now that we have moved into solving problems within 20, the students is able to draw the pictures to represent solving a word problem, but when they go to count and solve the student struggles with counting above ten with one to one correspondance. Reading this section about Emily made me think about what steps I now need to take with my student to help to build number sense above 10. Just like Emily, my student needs lots of work with counting above 10 in different ways to help make the connections needed to become proficient with solving problems above 10. Sometimes I feel like a rush through these essential number sense building activity because I want to get straight to learning to add but without allowing children to make connections through exploration it effects their ability to be proficient. I have noticed with my student that because I moved away from having them work on counting it is impacting their ability to even measure objects accurately because their one to one correspondance is not solid. Reading this chapter has helped me to examine at least one of my students and is helping me to plan next steps to help my student to become profient.
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  • 15 Jan 2018 3:58 PM | Anonymous member
    Students are asked to move 10 ones to the tens column when adding two digit numbers. We have used straws, colored chips, and diagrams on white boards. I saw students making connections with place value and addition, but not the whole class. As teachers, we need to be willing to use class time to develop connections for all students. This week the lessons include adding 3 digit numbers to 2 digit numbers. Students need to make the connection of place value before adding the numbers together. The sum has to make sense. Using the stories. hopefully will help.
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  • 16 Jan 2018 7:01 AM | Anonymous member
    I have a few Emilys who do much better with stories. I also have the opposite of Emily, those who can't visualize a picture in their minds when working through a math problem. I encourage my class to work through problems by drawing a picture then writing their number sentence, or writing their number sentence then drawing a picture to show this problem.. I encourage them to use what is easiest for them then ask that they step outside their comfort zone and try something different. We work in mixed groups so there is a go to person when they feel they need support.
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  • 16 Jan 2018 3:52 PM | Anonymous member
    if connections aren't made then they have no context of how to add or subtract with any accuracy. Students need to first understand what numbers represent before they can perform any calculations with them. Some students are good at memorizing facts and then struggle when they actually have to calculate using bigger numbers ,because they are missing that relationship of how numbers go together to form other numbers they have difficulty breaking numbers apart and putting numbers together. When they are missing those basic foundational skills math is difficult for them unless presented to them in a way they have made a connection to try and figure out the process. Trying to untangle these learners is often difficult when they sometimes are not sure where they get lost in their thinking. Getting them to explain their thinking is sometimes frustrating because they answer "I just know it" or "I thought it in my head". So their tangle is hard to uncover and try to help them untangle.
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  • 17 Jan 2018 8:11 PM | Anonymous member
    Students come to the classroom with a variety of experiences. I find that students need to have time to explore math concepts just as Kathy Richardson stated on page 198. I also feel it is important that as students are exploring we circulate and discuss their understanding.
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  • 17 Jan 2018 9:52 PM | Anonymous member
    This story made me think of a recent experience I had with a group of 7th graders. We spent about a week exploring with "hands on equations" as an introduction to solving one-step equations. The students discussed and worked through solving a variety of single step and multi-step equations by modeling. They were able to describe if something was being added, taken away, etc. They came up with the idea that in order to stay balanced/equal you had to do the same thing to both sides. The kids seemed pretty proficient they were able to model an equation, write the equation using numbers, and solve the equation by manipulating the pieces and describing the process.
    The first day I gave them "naked number" problems they were in an uproar. They had no clue how to approach the problem. I reminded them about all the work we had just done and asked them to close their eyes to picture the equations. Once they made the connection they were able to go through the problems with little difficulties. If a student was stuck I would always go back to help them make a connection to the model. After a few days of practice they were pretty confident.
    I was even able to challenge them with some equations like, solve for x if x + y = z and 2x + y = z. They were successful because they made connections and their minds were not focused on what was unknown.
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  • 17 Jan 2018 11:26 PM | Anonymous member
    In a high school special education math class that I have taught several times, students are much more likely to make sense of problems if they are put in a real-life context. These students can have more "gaps" and misconceptions in their math education. The students make sense of solutions most often when there is a diagram as well. When recently working with the Pythagorean theorem, the focus initially was on finding the length of the hypotenuse. If it had to be the longest side and a student got a value smaller than one of the leg lengths, I could work with their “common sense” to help them understand if there is a problem with calculations. Just making the calculations with no “reasonable result” has no meaning to the students.

    I am inspired to explore more opportunities like this as we begin a basic Right Triangle Trigonometry Unit. I might start with inquiry by labeling the sides of a right triangle O, A, & H (similar to an introduction measurement activity that I use to develop the ratios sin, cos, & tan). The I would ask the students to see how many fractions that they can make with the 3 letters. I might next give values to O, A, & H using a Pythagorean Triple (which we discussed in the previous unit). Next, I would have them explore which fractions are proper fractions and which are improper fractions (and have each student make some conjectures about why some of the fractions might always be proper - O/H sin and A/H cos). Doing this would help the students confirm their measurement results from the introduction activity. To add in a context, I would revisit some of the word problems that we worked on in the Pythagorean Theorem Unit - looking at some of the same problems in a different way (exploring how we might solve the problems given only one side length and two angles - 90 degrees and ?? degrees). “How would increasing or decreasing the reference angle affect the length of the side labelled “O” (or the distance from the building that the base of the ladder is placed)?

    Just giving the students the 3 Trig Ratios and applying them to right triangles would probably have little meaning and the ratios are easily confused. Visually selecting the opposite and adjacent sides can be a challenge itself.

    Pam
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  • 18 Jan 2018 2:20 PM | Anonymous member
    Overall, the students in my alternative math class are excited when they are given story problems, and they have become quite adept at creating visuals that depict the problem at hand. Most of the time they can talk through different strategies they could use to solve a particular problem then they get hung up when they start putting numbers on paper. Sometimes, it is like they hit a wall and their understanding gets eclipsed when they start to use the numbers given in the problem. Often it takes several probing questions to get them started down a path of reasoning out how to use the numbers with their problem solving strategy.
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  • 20 Jan 2018 11:54 AM | Anonymous member
    Wow. After reading about Emily's story, I wonder how many of my high school students know facts and don't really have a real connection to the arithmetic operations. Also, I don't know how many times I have shown kids different ways of doing something in hopes that one of the ways will make sense to them without taking the time to have them discover a connection to previous knowledge. I might have made the connection for them. They may or may not have understood the connection.
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  • 21 Jan 2018 8:56 AM | Anonymous member
    Emily's story is one that teacher's can learn from especially at the younger grades. When I think of my class many of the students are opposite to Emily because they give you the number answer but can not explain how they got it. I agree with many of the posters that giving our youngest learners to time to explore the concept in a guided way will give us the best benefits in helping all children learn the mathematical foundation needed for their entire school career.
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  • 21 Jan 2018 2:47 PM | Anonymous member
    I loved the story of Emily and how Tracy was able to have a conversation with her to see what her misconceptions were. I found it interesting that when there was a story, Emily was able to make connections and the problem had meaning for her. I sometimes see the opposite. In solving 8th grade math word problems, students are often turned off by "all the words" and don't even want to attempt the problem because it looks too hard. The reading sometimes gets in the way of the math. I would expect that students who have a stronger background in solving story problems wouldn't feel that way. It is so very important for elementary math classrooms to have the rich conversations and problem solving opportunities so that students can see the relationships and connections in math.
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  • 21 Jan 2018 6:01 PM | Anonymous member
    Interesting story! It feels like the practical story problem helped Emily to solve the problem despite not knowing basic facts. Having that meaningful connection to the story (experience + language) allowed her to understand the math vs. just the naked numbers. Building on this experience (and her use of language in math) to better understand symbols and math vocabulary could be very helpful for her. There are such great interconnections.
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  • 21 Jan 2018 9:12 PM | Anonymous member
    Emily so makes me think of one student in particular in my classroom. This student struggles daily. I work with her often one on one, but she lacks number sense. This child is so extremely shy to begin with, and will not ever ask for help. I have tried so many strategies to help the student learn addition of bigger numbers. If I help the student, we either have to use our fingers or manipulatives and count them, but it still seems like they are not making the connection. This student is a special education student. I can only imagine how frustrating it is for them, when I am frustrated because nothing seems to help them.
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  • 23 Jan 2018 4:47 PM | Anonymous member
    In reading Emily's story, I thought a lot about how we judge the extent to which students understand math. Many students are the opposite of Emily-- they can show on paper that 7 + 7 = 14, but they do not understand what that means! I found it fascinating that Emily could do the more complex word problem (something with which my own students struggle), but she incorrectly answered most of the rote addition paper. This also made me wonder about the amount of time we spend having students answer a million rote addition/subtraction/multiplication/division facts... I can see the importance of having math facts handy so a student can fluently do a problem "in their head", but I wonder about how often I misconstrue a student "getting it" when really they have no idea what the equation means!

    I am currently teaching order of operations to a multi-grade middle school group. Once again, I am amazed at how many of my students cannot do rote multiplication/division mentally-- didn't we spend YEARS giving them facts tests?! How can they not do this? Again, I am questioning the rote memorization technique that we have practiced for years.

    Yesterday, I had one student who skip counted by 9s to answer a problem-- a strategy that clearly has worked for him in the past, but a) was not very immediate, and b) he counted wrong and therefore ended up with the incorrect answer. This is a student who scores low average to average on standardized tests, and yet he was unable to complete the first part of a multi-operation equation.

    These two recent experiences have led me to see that I need to be very balanced about the amount of time I am having students spend on simply "spitting out" the answer. I need to be checking on their true understanding of the problem instead of simply checking to see if they have come up with the correct answer.
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  • 02 Feb 2018 6:45 AM | Anonymous member
    I see lack of connections being a problem daily.The program we use will present many different strategies to problem solve but insist that children use them. The problems not only are meaningless for the students but not all students relate to the strategy that is being tested. I try to reword the problem and personalize it when I see the children or individual child can't relate to it. Sometimes we role play the situation or put in the child's name in the problem so the situation is more relevant. This connection to the problem makes a huge difference in the attitude of the child and the ability to understand it and solve the problem. Also the more you bring in movement, drawings and objects to help with finding the solutions, the more engaged the students are because they can "see" the problem and work with things to solve it.
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  • 03 Feb 2018 10:24 AM | Anonymous member
    Chapter 8
    Reading Emily’s dialogue with her student made me think of my students. They want to keep change and flip, and they have no idea when or why or what to flip. I have banned them from saying such things, unless they can tell me why they are doing such things.
    I started jumping around the room doing 180 degree turns and they asked me why are you doing that, and I replied. "I don’t know someone taught me how." They rolled their eyes, but they started to get the point. I looked ridiculous and saying because someone taught me was a silly explanation.
    I tell them, don’t do something because you were taught. Ask yourself what does that mean? Why am I doing this? Does this make sense?
    For example: We are learning about one and two step equations.
    And, they are understanding that when they are dividing to undo multiplication, they are multiplying by the multiplicative inverse.
    Why?
    Because they want to end up with a 1, so that when they multiply it by the variable, they end up with the variable. (Multiplication Identity Property)
    When they are undoing addition or subtraction they want to have a 0 to add to the variable because of the Addition Identity Property.
    Not because someone taught them.
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  • 08 Feb 2018 4:33 PM | Anonymous member
    I teach special ed. I have found the opposite of Emily. My students learned a strategy that they have memorized to help them solve basic math problems. Therefore, when it comes to word problems my students won't even attempt them. They skip it or wait for help. I have been working with them using a step by step approach. I have been using the CUBE method. They still won't make the connection independently. It is taking a lot of practice.
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  • 09 Feb 2018 5:51 PM | Anonymous member
    I would say in a general sense that students don't necessarily see math topics as connected, and they also lack the confidence to look for and draw parallels.

    Sometimes when we present the connections explicitly, I can see "ah-hah!" moments, and "Oh! We can use THAT here?!" as if students were waiting for permission to use something. This was evident in using the reflexive property in triangle proofs recently. The idea that x=x is the reflexive property is abstract and plain weird in algebraic proofs, but voila! has a purpose in geometric proofs.

    What happens next, though, is beautiful, because once students have permission to make connections, they begin to look for them, and -- especially when the students find and make connections on their own -- they understand better.

    I do think I could do a better job of ASKING students what connections they see, even if I have to nudge them toward where to look, instead of connecting too many dots for them too soon.
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  • 16 Feb 2018 10:20 PM | Deleted user
    I believe that connections are so important! When I was teaching a special ed math class last year, my students were able to tackle quadratic word problems (write and solve equations). My colleague (Pam Coulling, participant in this group) and I discussed how contextualizing (word problems rather than naked problems) was the key to their success. The lack of connection affects students' proficiency. Two particular problems that I see, operations with fractions and working with signed numbers (major culprit is subtracting from a negative number, -2-5). I've tried to help students develop a story of their own to build understanding, by offering several examples.
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  • 20 Feb 2018 8:24 PM | Anonymous member
    When I am introducing a new topic or reviewing something that has already been covered in class, I try to make connections to the students in how they will use the math in the future. They always ask that question and I try to give examples whenever I can.

    I also try to have students share when they think they will use the math in their lives. I like hearing their thoughts and they often spark the interest of their peers. They have a lot of background knowledge and experiences that we need to connect with so they feel like they have contributed to the class and use their math knowledge.

    As I continue to work with students, I want to make sure they make a connection that helps them remember the math. Having students make up the math problems for their peers can help the kids make the connection with the math and also their peers make connections with the topics in the problems
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  • 25 Feb 2018 3:32 PM | Anonymous member
    I was reading through the comments others have made and I find myself in similar experiences where many of my fifth and sixth grade students get confused when their problems become word or story problems. Like others mentioned, these students have memorized some of the facts and steps but then unable to connect it to a real life situation. I have really been trying to incorporate "what strategy did you use" when we discuss our math problems. I find that there are multiple ways to solve and for students to hear this from each other is more powerful than me just telling them.
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