2.91+Reflection+10+Tech-Knowledgy

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Paste your reflection on "Tech-knowledgy and Diverse Learners" followed by your name. This is due Sunday 12:13 am.

When I was in school, the only form of technology I can think about in my middle school math classes was a calculator. Even then, I was the only student with a TI-84 and everyone else had a little one. No one knew how to work my calculator, not even myself, and not even my teacher. Everything we did was done on the board or by hand and nothing was given to the hands of technology. This is why I believe I have such a difficult time with technology, especially for academic purposes. Suh talks about the idea of “tech-knowledgy”. In her words, this is “the knowledge necessary to use cognitive tech tools effectively to construct mathematical knowledge, evaluate the mathematical opportunities presented, and design learning tasks with these tools that amplify the mathematics.”(2-3). Somewhere along the way, I think my middle school teachers lose their tech-knowledgy. Suh also talked about those groups of students that benefit most from the use of technology. First she talks about those who have English as a second language. She states that not only do these students have to learn English in math, but they also have to know it conversationally to work with it (2-5). Coming from someone who is also a foreign language major, I can see how technology could help. I know when I can see what I am working with in my Spanish classes, whether it be a story, text, or work sheet, I am able to understand it more. Visuals help students learn. For Once of my Spanish classes, we had exercises we did online with an interactive website and it helped. I know if I were trying to learn math in Spanish, I would definitely benefit from something interactive. The other group Sun talked about benefiting from was special education students or “low” students. Being able to work with technology constrains them so they are able to develop their physical thinking (3-4). I look at this information and think about what would have happened if I were given the opportunity to learn through the use of technology. I always considered myself on the higher end of learning and I wonder how much more I could of understood if I were allowed to have experienced the virtual manipulative. The experience Suh talked about with the online website which used tree growth to explain slope amazed me. The idea of the site was to have students learn through making conjectures about what if statements and think about how something would affect something else (5-6). When I first learned about slope I got all the algorithms right away and was able to do the algebra with it without a problem. It took me until high school calculus to truly understand what slope was. I always told people it was rise over run, a saying my school used. I had no idea it was really a rate of change and the vast amount of things we could apply it to. I was shocked when I found out. All in all, I think I could really benefit from utilizing technology with my students. My only thought is how do I learn about all of these technologies and how to I learn which are best for them to learn?

Katey Cook

In Toh’s article, “Tech-Knowledgy and Diverse Learners,” the author starts by praising technology in the classroom, as a way to amplify mathematical concepts. By “amplifying,” Toh implies that these cognitive technologies can do a lot to enrich our mathematics teaching and learning by providing vivid visuals, models, and graphs at a moment’s notice (2-2). More importantly, however, these tools allow us to “off-load routine tasks, such as computations, to compact information and provide greater efficiency in learning” (3-6). After first reading this I was reminded of the tediousness of endlessly graphing y=mx+b that I felt as a middle-schooler. Rather than using valuable class time creating grids and finding an appropriate scale, the students are instead spending time looking at the concepts directly applied to a relatable, and sympathetic, situation, as referenced with the “environmentalist” problem (5-5).

Toh pointed out the special application that these types of cognitive tools have for students with diverse learning styles. She mentions specifically those who are English Language Learners, as they are in a unique position of potentially being highly skilled with mathematics, but are not able to participate in most classroom discussions due to the language barrier. She cites a 2005 study stating that ELL’s in addition to lower ability level students benefit from using virtual manipulatives, saying that their built in constraints “can help over come some limitations of their physical counterparts” (3-4). I feel that these limitations must include the class time taken to set up any manipulatives and clean them up afterwards. By changing to a digital medium, no time is wasted by the distractions that can arise during these non-instructional periods.

One thing that did really catch my interest was Toh’s statement, “This example is not meant to discount the importance of students’ ability to translate data into different graphical forms” (6-5), while referring to off-loading the task of creating a linear graph by hand onto the technology. Though this allows for more time to be spent talking about the graphs themselves and what they actually represent, I think it is necessary that we realize that we are loosing something, also. As tedious as it is to make graph after graph, the computations helped me to learn simple patterns of various operations, and how to do quick mental math. I am not recommending that we focus on computations as opposed to concepts, but by cutting out a large portion of the computations that the students do, the less they will be skilled at concrete numerical operations. These critical skills, though not conceptually high-level, are more common to people on a day-to-day basis.

Valerie Gipper

The article Tech-Knowledgy and Diverse Learners by Jennifer M. Suh is an interesting analysis of the various types of technology that can benefit diverse learners like English Language Learners (ELLs) and students with special needs. The author begins the article with a great definition of her use of the hyphened word tech-knowledgy. She says tech-knowledgy is “the knowledge necessary to use cognitive tech tools effectively to construct mathematical knowledge, evaluate the mathematical opportunities presented, and design learning tasks with these tools that amplify the mathematics” (441-3). In the twenty first century I think it is important for us as educators to understand this tech-knowledgy and keep up to date in order to best prepare our students for the future.

I found it interesting that the author chose to analyze how tech tools can benefit students that generally struggle in mathematics classes: ELLs and students with special needs. I wonder why she doesn’t explain how these tech tools are universal tools that should be accessible to all students. As Suh put it, many of these students have, over time, learned helplessness due to negative experiences in mathematics (442-1). She goes on to explain how students with special needs usually are taught mathematical topics by direct instruction with repetition of the major procedures (442-2). Another strategy that we’ve discussed in class is the use of manipulatives and how they can be a more concrete, hands-on approach for visual learners. Suh agrees that manipulatives can be helpful, especially for ELLs, but they can be a bit overwhelming for many students with special needs (442-4). I’ve seen this first hand with one of the kids I babysit often. He has been diagnosed with Asperger’s and Sensory Processing Disorder. With his conditions, he has a hard time keeping calm when he is overloaded with too much information/new people/lights/sounds or all of the above. I can see how he could have a hard time focusing with the use of too many manipulatives in a classroom setting.

In the sub-section “Developing Tech-knowledgy for Diverse Learners” Suh mentions an idea brought up by Kaput who said that “tech tools in mathematics learning and teaching is the ability to off-load routine tasks, such as computations, to compact information and provide greater efficiency in learning” (442-6). In my pre-internship experience, I’ve seen this to be true. After learning some of the computations by hand my teacher has allowed his seventh graders use calculators for some harder versions of questions.

The author goes one to give three real-world examples. First, the Healthy Forest and how it applied to rate of change (444-1); second, Building Rules for Linear Functions (444-9); and third, Balancing Equations Using the Virtual Balance Scale (445-6). These three examples use various visuals to aid student learning and has “allowed students to focus more on the mathematical processes and relationships and make sense of the mathematics” (446-3).

After reading this, I think it would be great to use some of the online visual tools like the virtual balance in my future classroom.

Tori Ward

In” Tech-Knowledgy and Diverse Learners” author Jennifer Suh explores the use of technology in the classroom to engage diverse learners. Suh argues that these tech tools will “amplify” mathematics (1-2). These tools allow students to view representations of the math in different ways. Suh also talks about the need of teachers to evaluate the tech tools they use in their classroom to ensure that the tools are enhancing the learning rather than diminishing it (1-3).

The diverse learners she talks about engaging are English language learners, and special educations students. In regards to the special education students she mentions a “poor academic self-concept” and “a negative experience in the classroom” that create a learned helplessness (2-1). How do we reverse this process? We want to challenge these students and give them a successful classroom experience, and have success in their academics, but this is a very difficult challenge for teachers. Suh talks about students not being pushed to a critical thinking level (2-2). My question would be do these students want to be pushed to a critical thinking level? From my experience in Mr. Wine’s class the students didn’t like the focusing questions. This could be a trained helplessness that Suh talks about earlier.

There is an emphasis on developing technology for the diverse learners. The tech tools can provide students with visual representations, give immediate feedback, are interactive, and give an ease of differentiation according to Suh (2-7). Suh even created a resource to evaluate the effectiveness of tech tools within the reading (3). She talks about special needs students are given direct instruction on how to perform algorithmic procedures, without gaining the understanding of the concept (3-2). I think the best part of the article is when she gave examples of a few different real world problems. She mentioned a website that showed tree growth in a forest (4-2). The students can see the different rates of change immediately through the graphs, and with understanding the labels of the graph they will be able to say which tree grows at the fastest rate. I think the best example was number two. With the three different MP3 plans students can evaluate which is the better buy for them (5-1). I feel that many students that I encounter at the schools I’ve subbed at can relate to this problem because I’ve seen many of them with IPods and other MP3 playing devices.

Overall I think Suh makes some good points about using technology in the classroom to help students make connections and amplify mathematics. I think the technology can only amplify mathematics if the teacher knows exactly what they want to accomplish using the technology. Teachers need to evaluate the technology they use and make sure that it enhances the learning rather than creating extra confusion. Technology should be implemented in the classroom with careful thought. I feel that technology can create a lot of problems in the classroom. Technology doesn’t always cooperate the way we want it to and teachers need to have back-up plans in case there is a failure of technology in the classroom. Students have to be able to understand the technology and how it is making the different visual representations to successfully make connections to the math.

Mike Freeland

===Technology can be very a beneficial tool in the classroom. It can be employed as any other manipulative to help demonstrate an abstract concept in a concrete configuration (2-3). "Techknowledgy" is the knowledge to use tools effectively to amplify mathematics. Techknowledgy can be developed to enrich learning for a diverse body of learners. There is a wide variety of applications for technology: linked representations, graphing, collaborative learning, linked vocab, guided exploration, immediate feedback, online calculation aid, tiered lessons, language switchability, customizable and replaceable (3-1).=== Students given technology to aid in tasks found that the real-world application and ease of use made it easier to engage. They even seemed to not realize the math they were doing, but rather, their minds were being stimulated and they exercised computation methods as well as problem solving (4-5). Technology aided in giving some students the graphs right away rather then asking them to construct it themselves. This differentiation method allowed for multiple tiers of students to participate in the same activity and collaborate for discussion and collective learning.

kyle d.

In this article, Jennifer Suh focuses on the capabilities of technology-based learning. She states, “Us­ers of these tools have the capability to graph, model, compute, visualize, simulate, and manipulate, which //amplify// mathematical properties and concepts.” (2-2) Suh also focuses on two specific groups of learners, both English Language Learners (ELLs) and students with special needs. Those who are learning English will have to gain an understanding of the mathematical concepts and it can be difficult to learn to talk about mathematics. (2-5) With special needs students, research has shown that they can have difficulty grasping metacognitive thinking related to mathematical problems. (2-6) Usually, teachers focus on direct instruction with these children, but more research shows that special needs students can learn using other methods. (3-1) There is then talk of the importance of teachers knowing how to utilize knowledge in their classroom. (3-5) Also, to judge how to get the most out of a certain tool, Suh created a list that allows teacher to see what is valuable in a tool. (3-6) This is an effective list because there are many different categories and how each tool addresses the needs of ELLs and special needs students. It is interesting to look at the examples giving throughout the rest of the reading. In her first example, Suh wanted her class to understand graphs and how they show different rates of change. (5-2) This was done by having the student being able to manipulate the age of different trees and having a graph to show how the trees grow depending on how old the tree is. I find this to be an effective way to show a real world example of what they see in the graphs. As a result, students are able to identify trees on size at certain points, when trees grow the fastest, and other important mathematical ideas. This is continued in the next example, but this time with different contracts for downloading music. (6-1) This gives them experience comparing and contrasting rates with something relatable. It also saved students time where they could have plotted points themselves and lessened the likeliness of error by graphing. I believe the effectiveness of tools has been fully justified by the examples given throughout the article. While the focus of this article is greater on those who have a language barrier or other special needs, it should be just as effective for other students. These examples not only give students a chance to explore on their own, they allow them to get away from the sometimes less than exciting worksheets and lectures. The benefits of using technology are too great to not be utilized and a teacher is doing a disservice by not utilizing the technology we have in this day and age.

-Marcus Edgette

I had a bad feeling going into this reading, mainly because whenever I hear “technology in the classroom” my brain dismisses whatever comes next. I figure my best and worst case scenarios: my worst case scenario (which seems more reasonable for new teachers) is me in a poor school district, managing a classroom where all I have is the computer on my desk, and a chalkboard with a few already used sticks of chalk at my disposal. Best case: All my students are provided with I-Pads and I’m up front with a “Smart Board”, and who knows what else. My philosophy thus far has been to hope for the best, but prepare for the worst. So I never figured studying available teaching-technologies would be worth my time, not until reading this article.

I always thought teachers used technology to make //their// lives easier: less instruction, easier grading procedures, quieter classroom… but author Jenifer Suh makes it clear early that a teacher who can effectively use computer applications can address the varying strengths and needs of different students. The objective of “Tech-Knowledgy & Diverse Learners” is to leverage the power of technology to teach math and engage diverse learners in problem solving and critical thinking (441-1). These are not the traditional “drill-and-practice” computer programs some of us may have grown up with. Tools today can graph, model, compute, visualize, simulate, and manipulate; all aiming to amplify mathematical properties and concepts (441-2). Teachers are responsible for a range of diverse learners; this technology can support them, but only if teachers have the knowledge necessary to use such tools effectively: “to construct math knowledge, evaluate such knowledge, and design learning tasks accordingly.” (441-3). At this point in the reading (still on page one) my mind is now open, intrigued even, to how some teachers are using what forms of technology in their classes. Maybe I //could// use technology, even in my worst-case scenario. NEEDS OF DIVERSE LEARNERS

The diverse learners this article focuses on are English language learners (ELLs), and students with special needs. I have studied both learners in my SPED 4290 class, but back then it seemed like making accommodations for these students often required extra time during lesson planning, and sometimes meant extra resources/assignments, approaches of teaching… a multitude of modifications. Suh suggests that technology could simplify the process of making accommodations.

ELLs often require assistance with the language in their math class; both conversational and academic languages (441-5). Special education students, on the other hand, struggle with the metacognitive aspect of problem solving, like selecting appropriate strategies, organizing information, and generalizing new strategies. In class we talk about how the use of physical manipulatives, like patter blocks or algebra tiles, could help, but there are pros and cons to those as well. Suh claims that, “When students work with manipulatives, one major challenge is that handling multiple pieces can create an excessive cognitive load for learners… Essentially, students are unable to track all their actions when using the manipulatives. They also struggle to connect multiple actions with mathematical abstraction and symbol manipulation.” (442-3) Virtual manipulatives can help overcome some of the limitations of their physical counterparts. DEVELOPING TECH-KNOWLEDGY FOR DIVERSE LEARNERS

One impact of tech tools in math learning/teaching I quarreled with was its ability to “off-load” certain routine tasks, such as computation. I feel like computation should become a mental reflex when approaching a problem and calculators take away from the thought process in deriving answers. I see students in my middle school pre-intern classroom who struggle adding and multiplying without a calculator! I grew up hearing (and used to say myself) “You won’t always have a calculator in the real world…” but today basically ANY phone can take care of any reasonable calculation one may come across day to day, so perhaps calculations DO get in the way, or complicate learning new material. So what I thought was a drawback to relying on technology isn’t so bad after all, and Suh, Johnston, and Doud’s provide a list of five benefits of math tech tools (442-4):

1. Linked representations can provide connections and visualization between numeric and visual representations. **2. Immediate feedback allows stu­dents to check their understanding throughout the learning process, which prevents misconceptions. ** I like this one a lot because sometimes students think they learned something, and then practice that misconception over and over and over again, until the incorrect process is engraved into a procedure. Immediate assessment will correct any misconceived learning right away, so every student learns something right the FIRST time, and later teachers won’t need to stop what they’re doing to make sure everyone’s on the same page. 3. (Didn’t like this one) <span style="display: block; font-family: arial,sans-serif; font-size: 16px; text-align: justify;">4. Opportunities to teach and represent mathematical ideas in nontraditional ways. **<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">5. Ease of differentiation and scaf­folding help meet the needs of diverse learners and others. **<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">I also “bolded” this one because this would dramatically simplify lesson planning for classrooms with special needs students. I imagine a math program in class where all the students are learning the same thing, but it is presented slightly different for varying levels of intelligence, so that lower level learners could perform more simple tasks while higher level learners are properly challenged by the same program at a different degree of difficulty.

<span style="display: block; font-family: arial,sans-serif; font-size: 16px; text-align: justify;">EXAMPLES OF COGNITIVE TECH TOOLS IN USE

<span style="display: block; font-family: arial,sans-serif; font-size: 16px; text-align: justify;">According to Suh, “Conventional instruction on algebra tends to be rule based… (and) special needs learners often receive direct instruc­tion on how to perform algorithmic procedures... without having opportunities to construct conceptual understandings of mathematical pro­cesses.” This is a common reason that students have a hard time recalling what they have learned in previous years, because they forget particular rules and when they apply. It seems difficult to teach math without laying down “rules” students must follow, but examples of tools used in real middle school classrooms sounded very neat.

<span style="display: block; font-family: arial,sans-serif; font-size: 16px; text-align: justify;">1. __Healthy Forest__ “applet” (brainingcamp.com): gives students opportunities to explore and develop a sense of the mathematics (444-1). While reading about this program I thought to myself, “Aw, this sounds like fun! And it’s teaching kids about rate of change, graphs, functions, problem solving, and connections? Why couldn’t I learn that way?” It looks very interesting, and apparently the students involved felt the same way I do. <span style="display: block; font-family: arial,sans-serif; font-size: 16px; text-align: justify;">2. MP3 plan problem for __Linear Functions__. Fun and relatable way to visualize and compare linear functions! Spreadsheet is provided for special needs students, to allow them more time to analyze different plans instead of spending time plotting points. (444-7). <span style="display: block; font-family: arial,sans-serif; font-size: 16px; text-align: justify;">3. __Balancing Equations__ Using Virtual Balance Scale (http://nlvm.usu.edu): records the process of balancing algebraic equations. (445-4) This is basically the way I still envision solving variable expressions! By making sure the equation is “balanced” on an old fashion scale. Very cool.

<span style="display: block; font-family: arial,sans-serif; font-size: 16px; text-align: justify;">FINAL THOUGHTS

<span style="display: block; font-family: arial,sans-serif; font-size: 16px; text-align: justify;">This article has opened my eyes to the world of teaching technology, and I am anxious to continue looking. Although I still see technology in the classroom as a luxury, it can be a service to both the teacher //as well as// the students in the classroom utilizing it. Tech tools eliminate the task of comprehending both pictorial image and symbolic notations as they change in response to the students’ input, allowing students to focus more on the mathematical relationships and processes while making sense of math (446-3). As Suh puts it, “These structures free the student to focus on the connections between the actions on the two systems [notation and visu­als] actions, which otherwise have a tendency to consume all of the stu­dents’ cognitive resources even before translation can be carried out.” This explains why students in the video we watched, of Cathy’s class, could invert and multiply, but could not describe how to divide a fraction. <span style="display: block; font-family: arial,sans-serif; font-size: 16px; text-align: justify;">“Effectively teaching math today will require teachers to consider the best available resources, based on research, and discriminate/evaluate these resources so that they can be implemented to yield the most meaningful learning for students.” “That” is Tech-knowledgy.

<span style="display: block; font-family: arial,sans-serif; font-size: 16px; text-align: justify;">- Tim H.

The authors of this article stress the importance of “cognitive mathematical tech tools” in the middle school classroom, such as computer programs (441-2). The goal in using these is to engage students with diverse learning styles in problem solving and critical thinking (441-4) and to “amplify” the mathematics (441-3). According to the article, teachers must have “tech-knowledgy”, the knowledge and ability to incorporate technological learning tools into the classroom (441-3). Some advantageous aspects of these tools are that they can help with connections and visualization, give immediate feedback, and promote interactivity (442-4).

The authors emphasize two groups of learners that might especially benefit from this: English language learners (441-5) and special needs students (441-6). There was quite a bit of information in this article regarding special needs students and how to relate technology to them. One important point was that often a poor academic self-concept combined with negative mathematical experiences contributes to the learned helplessness of these students (441-6). We saw this in Mr. Wine’s class with a student who insisted that she just wasn’t “good at math”. Interestingly, this “tech-knowledgy” can actually be more manageable for special needs students than physical manipulatives, since it doesn’t overload them cognitively (442-2). The speed with which calculations are made using technology tools allows students to spend more time understanding mathematical connections rather than performing routine operations (445-1).

There was relatively little information in the article regarding ELLs. I think it is a great asset that some online resources can be translated into another language, but I wish the author(s) had said more about these students. As a Spanish major, the language learning aspect of both schooling and mathematics particularly interests me. With our next school visit, there are immediate and helpful applications for kids with special needs, but what happens when we get nonnative English speakers in our classrooms that are confused? Last year, I worked at Lake Center Elementary School as a parapro/ELL tutor for a Latino student. He seemed to be very good at understanding mathematical concepts, but language was definitely a barrier. As the book //Learning a New Land// points out, academic English takes 7 to 10 years to learn, which means mathematical vocabulary may be slow in developing for ELLs, and “tech-knowledgy” could help.

I think there is definitely a significant place for technology in the classroom. Over the past decade or two we have seen an increase in our overall use of technology, so this should logically carry over into the classroom. Right now, I do not have much experience using technology in the middle school classroom, but I have heard from my dad that short video clips and interactive websites can be extremely helpful and effective for making mathematical connections. The authors provided a helpful table of online resources (443), which is a very useful tool for current and future teachers. Most of these are free and easily accessible to students, too. With the increasing availability of the Internet almost everywhere we go, this is a wonderful way to connect mathematics and technology, which many students use in their everyday lives. I hope to greatly improve in this area and incorporate relevant technology in my classroom to deepen the understanding of my students.

Mandi Mills

“Tech-Knowledgy and Diverse Learners,” opens by discussing “cognitive mathematical tech tools,” which are defined as, “tools that are not simply the traditional remedial drill-and-practice computer programs. Users of these tools have the capability to graph, model, compute, visualize, simulate, and manipulate, which amplify mathematical properties and concepts” (441-2). These tools however, may take some getting used to for teachers as they need to take into consideration the differences of their learners in their classroom. In short, teachers need to learn how to make these tools a valuable asset to all their students.

Students such as English language learners (ELLs) or special education students may require a bit more attention according to the article. For example with ELLs, “the challenge is both the conversational and academic language that they need to use to participate in class discussions,” (441-5). On the other hand with special education students, seem to struggle the most with problem solving, such as where to begin and what strategies to use. This leads to the question, how do we as teachers learn to make these tools valuable for all the different types of learners that will be in our classrooms?

In the section, //Developing Tech-Knowledgy for Diverse Learners,// the article attempts to answer this question, with a list of 5 abilities of tech tools when they are used in mathematics, which include, “(1) Linked representations can provide connections and visualization between numeric and visual representations, (2) Immediate feedback allows students to check their understanding throughout the learning process, which prevents misconceptions, (3) Interactive and dynamic objects take mathematics from a noun to a verb, (4) Opportunities are possible to teach and represent mathematical ideas in nontraditional ways, and (5) Ease of differentiation and scaffolding help meet the needs of diverse learners and other,” (442-5). One of the things I think is very important is immediate feedback, such as with the calculators and the walking exercises we did in class. This way students are still involved with the problem, whereas with tests students do not get the immediate feedback, instead they have to wait and then once they receive their tests back they have to put themselves back into a problem and try to figure out where it went wrong. But with these tools students are getting immediate feedback and can see for themselves almost instantaneously what they did wrong and what they can do to fix it. Their minds never stop working on the problem.

I really liked the table in the article (443-1). I think it is something useful I could use in the future if I am struggling with ways to reach and help students with diverse needs. For example, the table talks about how immediate feedback can help students to not develop misconceptions and how it can also help students assess their own learning, which I think is important. I would like to be as hands off as possible. I want my students to learn from each other and in their own ways.

The article ends with, //Examples of Cognitive Tech Tools in Use,// which discusses three examples of lessons that were taught to students. These include, analyzing rates of change, building rules for linear functions, and balancing equations using the virtual balance scale (444-1). Example 1: Analyzing rates of change involved “Healthy Forest” an application involving real world problems where students can manipulate the data and see the changes it makes on the graph. Example 2: Building Rules for Linear Functions was about a proposed task in which the special education students to have more time to look at and think about the graphs in order to “analyze the different plans instead of spending their time plotting points,” (445-1). Finally, Example 3: Balancing Equations Using the Virtual Balance Scale. Here the students used a computer program to solve for x in linear equations by balances x boxes with 1 boxes. I think this was one of my favorites and could see how it could be beneficial to students because it gives them another way to look at a problem that cannot be portrayed on a white board. Kaitlin Froehlke

The article “Tech-Knowledgy and Diverse Learners” by Jennifer M. Suh is very appropriate considering next weeks circumstances. Before reading this article, I was a little apprehensive on using technology with Mr. Wine’s second hour class. They didn’t seem to use the manipulatives to their advantage, will they be able to use technology? After reading it, I am still a little frightened. I feel that the students represented in this article may have a stronger technological background than the students we are working with. However, I am maintaining high expectations for my students. When used correctly, technology amplifies and better assists student learning (441-2). This falls on the teacher’s “tech-knowledgy” to properly benefit the students (441-3). So, I have to ask myself, how far will I go so these students understand math?

According to the article, special needs learners receive more direct instruction (442-2). I feel that we shouldn’t lower our expectations or result to direct instruction simply because it’s easier. Special needs learners can achieve high expectations and reach an understanding of mathematics on a much deeper level than algorithms. To stray away from this, teachers have to do their research and try different teaching methods. Manipulatives can be used, but sometimes students struggle with making the mathematical connections (442-4). I noticed this in second hour. The students were not quite grasping the use of manipulatives with fractions and wanted to resort back to a procedural approach.

I believe the most important ability of tech tools, relating to our seventh grade classroom, is the immediate feedback that technology allows (442-5). Our second hour students were very focused on if their answer was right and didn’t really care about understanding the process. With technology, students can check their understanding throughout the lesson and make sure they are on the right track. They will not need to ask the teacher after every problem if their answer is correct, because they will be able to monitor with technology.

In the rates of change example, the Healthy Forest simulation provides students with a real-world scenario, and the program allows students to manipulate graphs and see how the trees change over time (444-2). Programs, such as this and the CBR, give students the opportunity to explore mathematics and develop a sense of understanding that cannot be achieved through algorithms. Slope can be confusing for middle-school students, but by using technology connections can be made about its importance.

In our classrooms, we need to use technology to our advantage and tie it into our curriculum. With so much emphasis on teaching to the test, it seems there is not much time for critical thinking problems (445-5). But if we provide students with opportunities to work with technology, they will have a better mathematical understanding and be required to use their critical thinking skills. I believe this will only benefit students in the long run and help them on standardized tests.

Hailey McDonell

While reading through "Tech-Knowledgy and Diverse Learners" I couldn't get out of my head something from the second paragraph. "By definition these cognitive mathematical tech tools are not simply the traditional remedial drill-and-practice computer programs. Users of these tools have the ability to graph, model, compute, visualize, simulate, and manipulate, which //amplify// mathematical properties and concepts." (441-2) This statement bothered me because it was stated so matter-of-factly when I don't necessarily think it's true. I feel like it's very possible that a student can know how to use math tools to graph, model, computer, visualize, etc without it amplifying any concept or property, and in fact, can have the opposite affect. I have discussed this somewhat at length with a friend of mine, how one of the problems a lot of students have when they get to the upper levels in math is the inability to disassociate a graph from a function. This sentence also made the rest of the reading feel like an extremely biased one, and while almost any piece of research has a specific goal in mind I felt this particular statement made the entire thing more difficult because of how quickly it was stated. It may not be a huge deal in general but it really turned me off and gave the rest of a it a "grain of salt" feeling for me.

One thing I did like about the article was it's focus on students who have difficulties with math, as I agree that this is where technology, if applied properly, can be extremely helpful in learning math. "For many of these students, a poor academic self-concept coupled with a negative experience in mathematics can contribute to learned helplessness." (442-1) That quote was very powerful, because it implies strongly that someone who has struggled with math isn't necessarily going to be horrible at it forever, they can still learn, it just might take a different approach. Various forms of technology, to help students visualize and analyze problems in a different light could be extremely helpful to these types of students. "...research has shown that using virtual manipulatives can benefit both ELLs and lower ability learners." (442-4)

Another good thing this article focuses on is that the teacher needs to know //how// to properly use the tools they have available to them to help the students. If the teacher isn't fully aware of how to use a specific tool it will likely end up being a waste of time, but even if they're an expert, if they don't know how they're going to go about connecting back what the tools do to the math it all ends up being wasted efforts anyways. Table-1 on page 443 was very good at breaking down //how// various things can be helpful to students.

Of the three examples I'd like to focus on Example 3: Balancing Equations Using the Virtual Balance Scale, as we are doing something similar to Example 1 in class and I didn't find example 2 all that interesting. "One feature of the virtual balance scale was that it explicitly linked a dynamic balance scale with the symbolic representation of the algebraic equations that were presented on the scale." (445-6) This particular tool, more than any of them, seemed to be amazing at connecting the math to the application. The fact that is would record everything the students did and //also// follow //typed commands// is rather fantastic. Not only will it tell students what they did when they physically move stuff around but it will show them exactly what they //think// they should do will accomplish. This example addresses a problem that I think most students face with things like algebra, which is the incredibly abstract nature of it. It's very difficult to get a concrete understanding of what's going on, and without that it's almost impossible to grasp the abstract concept itself. This tool seems like a fantastic way to help bridge that problem. Of the three examples it's easily the most impressive sounding and I'd love to be able to play with something like that some day.

While I had a rather sour taste in my mouth after that second paragraph I still came away relatively impressed with some things in the article. Some of the insights on how and to whom these technologies would be useful in the classroom I couldn't agree more with and as I said that third example really impressed me.

Doug Wills

The author of “Tech-Knowledgy” //; Jennifer M. Sue,// talks about how, “teachers must have ,tech-knowledgy: The knowledge necessary to use cognitive tech tools effectively to construct mathematical knowledge, evaluate the mathematical oppurtunities presented, and design learning tasks with these tools that amplify mathematics.” (441-3). It is with this knowledge where teachers can apply this method to all different kinds of students so that they can better understand mathematics. Through my personal experience I’ve noticed that the use of technology helped me better understand mathematical relationships. When I was in Middle school I can remember me having a T1-83 while everyone else had the regular non graphing calculator. I was noticing relationships amongst equations and the graphs real easily compared to my peers. The Article mentions that tech tools promotes critical thinking and problem solving. It also mentions the benefits it promotes to English Language Learners and Special Education Students. I feel that through the use of these tools it should also help out these students. With the different kinds of different instruction special needs students require, it is important to use things these students see in real life figures. We must be careful with distinguish between these real life figures and that’s where “Virtual Manipulatives”, comes in to play which students have a visual tool which gives them access to the mathematical language for them to talk in classroom discussions. Having the ability to use mathematiccal vocab not only shows you know what your talking about but it also helps other students realte to wahts going on.

As teachers become better integrated with the use of technology in middle schools teachers can impact the classroom in “providing greater efficiency in learning”(442-4). Some of the impacts listed as abilities of tech tools provided in the article are “Linked representations, Immediate Feedback, Interactive and dynamic objects, Opportunities to teach in different way, Ease off scalfoding” (442-5). I feel that these abilities are important in the classroom. It measures the accurateness of how well the student understands the problem. It helps them recognize problems quickly so the learning process is accelerated. I feel that the immediate feedback ability is the most important. It is within this ability where you can see how much of the learning process is being attained. It also helps out with fixing a problem a student may haave from the start and xorreecting the students error from a early state. The article lists 3 examples that show how technology can be used to solve problems. With Graphs, Models and spreadsheets you can “promote mathematical talk and critical thinking” (445-fig2). amongst students to help them further advance in the math class. These different examples all show how these additives further mathematical thought by relating equations to pictures. There stusents can specifiacally point out what they need to find the answer and can explain whats goin on in the picture.

In overall, I feel as we as educators should stay hip to the latest trends in technology that increase our chance to amplify mathematical thought in our students. Using the technology makes things much easier to grasp and easier to talk about what’s going with the tools amongst students. students can also take these tools home with them if its applicable to do so or they may ourschase the tools at their own leisure.

Fredrick Martin

In this paper I think that they pointed out how students can benefit from technology and the knowledge teachers can impart through this new medium. In the first part of the article Suh points out many difficulties students who have special learning need and are classified as ELL face. Some examples are first an excessive cognitive load that comes from handling multiple pieces (2-3), not making the connections between multiple actions and mathematical abstraction and symbol manipulation (2-4). Suh gives us a few good reasons why technology can be useful and how they help. The examples that were chosen were appropriate because they illustrate what can go wrong and what may be going right. In the first one the teacher notes that they often got caught up in collecting data which was not supposed to be the focus however they did try to make sense of the graph without knowing it (5-4). In the third example she showed how students with special needs did not waste all their cognitive ability on procedural activities and were able to instead use their cognitive ability on making connections when they manipulated the spreadsheet that was connected to the graph and from there think deeply about why when they changed one symbolic representation would it change the graph (6-5). I hadn’t ever thought about physical and virtual representations in this manner before. I always assumed that physical manipulatives were better for students because it helped them understand a little better. But now I can see the limitations as mentioned in the article, such as not being able to keep track of the steps you’ve taken and if not having to spend a lot of time and energy into graphing and reproducing what we did with the manipulatives or graphs in order to make the connections. I am glad that there are tools that we can use to help students understand what math is really about. Denise