Individualized Math Strategies

Problem being examined: Join Change Unknown:

student name finds 12 rocks on the playground. Everyone comes and wants to play with his rocks. How many more rocks does he need to find to have 21 rocks altogether?

This problem was presented to my small group of math students that I am working with this semester in my internship. It was one of two problems they were to solve during their independent practice. The problem was glued on a large sheet of construction paper, that would act as their, “works space,” providing them plenty of room to demonstrate a strategy that works best for them. One or two of my students occupy just a small area to solve their word problems, but for some, this piece of construction paper becomes an incredible space for them to break the problem down, or a map to represent the numbers using multiple strategies and trials, and more specifically, a space to solve!

One little girl, “Madeline,” amazed me by turning in a true treasure map of strategies that would eventually lead her to the gold (correct answer). I was sitting next to her watching this work unfold. Her first stop lead her to a traditional multiplication algorithm (which just happened to be something they were working on in class). Instead of letting her continue to solve the word problem this way, I asked her why she was choosing to solve the problem using the, “window” multiplication method. She responded that she thought that is how I wanted her to solve the problem. I gave her a huge smile, and reassured her that this is her work, and SHE should choose a way to solve the word problem that made sense to the problem itself and to her. She quickly crossed this algorithm out, and moved to an “inventive” direct model strategy. She began by drawing one stick person (which I assumed to be the student in the problem), and then started drawing five more people, stopped, looked at the problem again and made a self-prompted self-correction. She crossed out the

sentence, “Everyone comes and wants to play with his rocks.” Then crossed out her picture. She moved right along her treasure map, and started drawing circles. Madeline again did not resort to the algorithm, which I at that time noticed one of her peers was writing (21 – 12 = 9). She drew almost a diagram of circles, drawing lines to separate sections, and then writing the numbers each group represented beside them. I noticed then that she got frustrated when she realized she drew too many circles, but I at that time did not realize what that diagram was being drawn to represent. She crossed out that picture and moved to her final picture. She first drew 12 rocks and wrote the number beside them. She drew a line (similar to her diagram above) and counted up to twenty-one drawing one rock as she said each number. She finished by counting the rocks below the line and writing, “9 rocks.” This again represents a direct model utilizing counting up by one to solve for the answer. While she did not use the first diagram of circles, after looking at her work, I realized that incidentally she had indeed solved for the correct answer using that

diagram as well!

While this student is still in the first stage of solving word problems using direct modeling, as most of my students are, I feel through her self-corrections, and her ability to pair the whole numbers in the word problem with her pictures, she is on the brink of utilizing more inventive strategies to solve whole-number computation. She also drew her objects within a diagram-like formation, which is not demonstrated by her other peers, and is outside the norm of direct modeling. Her counters were drawn a way that represents the steps and what she needs to accomplish in order to solve for the answer. I feel when her confidence in solving increases, she will begin to develop strategies that break the problem down and demonstrate a solid understanding of the operation and its properties.

One method Madeline could have used to solve for this word problem, utilizing an invented strategy, would be to use her knowledge of whole numbers and subtraction. The student could take away the one from 21, which gives me 20 (starting with the largest whole number) and then mentally recall that 20 – 12 is 8, adding on that missing one gives me 9 so I know the answer is 9 rocks. This combines mental math with whole number knowledge. Another inventive strategy that would combine her use of mental math, while also utilizing a more manipulative-based way of solving, (using her direct model background knowledge) would be to break the numbers down using base-ten concepts. Starting again with the larger number, 21, the student would represent the number using two ten sticks and one unit square. The smaller number, 12, would be represented with one ten stick and two unit squares. This usesthe direct model concept, but instead of manipulatives representing the objects and the action, they represent the numbers. If the student does not have base-ten manipulatives, a picture can be draw instead. The student would first draw each whole number using the base-ten concept. The student would draw the two sides as separate, but then solve by combining, and taking away evenly to see how many were left. First a ten stick from each side is taken away (the take away has to be even), and then break apart the second ten stick into units. Now there are two units left from the 12 side and eleven units left from the 21 side. I take away my even sets again, removing two units from each side. The answer will be the number of units that are left.

Math Talk Moves

Today in my internship I observed a math lesson, with the focus on multiplying double-digit numbers. The teacher decided ahead of time that she wanted to teach a specific strategy to the students to help them solve this otherwise,“daunting skill,” in a clean and efficient way. I will call this strategy, “Box multiplication,” for my own and your clarification/benefit. The strategy stems off of place value, which just happened to be a previously acquired skill the students had just succeeded with the week before. It requires students to break the double-digit numbers down into two values: the tens whole value and the ones whole value (i.e . 42 becomes 40 and 2). The multiplication sentence is then re-written around a box with one inner box for each value being multiplied.

Example:

This was a strategy that was unfamiliar to the students, therefore today's lesson was spent really breaking down the strategy and it’s steps, and allowingthe students to practice it’s implementation many times. The focus was less on mathematical thinking and more onthe doing. I found three forms of Talk Moves, as discussed by Chapin, to be demonstrated in the lesson: revoicing, prompting for further participation, and asking students to apply their own reasoning to someone else’s reasoning. The first form of Talk Move that was used by the teacher was revoicing. This form of restatement was used most frequently in the lesson. In this context, the teacher affirmed and organized what the student said by re-stating the response with an undertone of clarification. (i.e. “You told me this (pointing to number) was worth ninety.” – student affirmed) I felt this Talk Move was very effective within this lesson, as it resulted in students’ having to confirm and affirm what their answer was. It also provided a source of repetition which aides in retention of new concepts for the student being revoiced as well as their peers. Students demonstrated their responses with the use of whiteboards while verbally responding to the teacher’s questions. The teacher also involved prompting students to engage in further participation. This was demonstrated by asking the students to provide additional information to what their peer had just said (i.e. completing the next step and then asking if the student agreed or disagreed with the answer a peer provided). Through this Talk Move I saw two different outcomes. The students struggled with being able to expand on their peer’s answer, and were often caught off guard by the teacher’s prompt. I felt that this was a result of the students not following along or answering the teacher’s question for themselves when their peer was called on to provide a response. On the other hand, I felt including this Talk Move repeatedly helped keep the students engaged and attended to the multiplication problem, even when they were not the one called on to start it. The more the teacher includes this form of extension into their lessons the moreeffective it will be in fostering deeper mathematical thinking and discussion. The last Talk Move prevalent in this lesson was the teacher’s attempts to get the students to apply their own reasoning to someone else’s reasoning. A great demonstration of this Talk Move was seen at the beginning of the lesson when a student was asked to explain why there were only two boxes used when setting up a two digit multiplied by a one digit math fact. The teacher then asked another student to use what his peer had just said to figure out how many boxesthey will need for math sentences with two digit numbers multiplied by another two digit number. This Talk Move was very successful within this context, and resulted in a class vote requiring students to agree or disagree with their peer’s reasoning. This required students to not only process the reasoning their peer presented, but also determine if they felt that reasoning to be true or not. Higher-level thinking was definitely occurring during this moment!

Upon reflection of this lesson I felt that while this specific learning opportunity had a lot of positive outcomes, there werealso some connections and thinking that could have really been enhanced by additional Talk Moves and questioning. Throughout the lesson I took note of a lot of, “What,” questionsbeing addressed to the student. These types of convergent questions only require one type of response, and the teacher is asking them to gain a specific correct answer. I thought through the use of more divergent questioning, students would engage in deeper mathematical thinking and solidify the strategy they were learning. Answering what 60 X 3 is requires a simpler type of thinking versus explaining why when the problemsays 65 X 3 we use 60 as a factor yet still multiply it by 3. When a student adds an extra zero or one to the end of their answer, instead of telling them that it is incorrect, I would ask them why they chose to add that zero, and clarify their reasoning so they may better understand the path they took resulting in the error. Having the students talk about their thinking out loud aides in retention and often helps the student self-correct and monitor where the error occurred. The teacher could really utilize restating and revoicing when having students explain their steps and reasoning for errors. Hearing their response or reasoning in a different way (by the teacher or peer) may help the student better understand what they were trying to express OR help them acknowledge errors and move in the direction of correcting them. Thislesson could really be transformed into an opportunity for students to, “talk out,” the multiplication problems they encounter. Hearing their own ideas expanded on through the words of someone else may present different ways to think about a problem, helping them solve a similar problem later faster and more efficiently! An additional activity could be for students to share their work and steps taken to solve the problems with a partner. This would be done after they solved the problem on their whiteboards (as what was done in today’s lesson). Once again, saying the steps out loud and explaining their work to someone else will ideally foster deeper retention and understanding of the strategy. Often these forms of interactions and discussions need to be formally or informally facilitated by the teacher, especially in math where there is a history of silent solving and written expression. Wait time would be a valuable component to include with these open-ended questions and discussion. Students require time to thinkthrough what they did, what they encountered and to be able to express what was written verbally. Often students feel rushed to answer a question quickly when in front of their peers. This pressure may result in error, leaving the student feeling more confused than before. Giving the students sufficient time allows them to dive in to their working memory and bring to the surface ideas and perspective used while solving the problem. I feel my CT already does an incredible job giving the students plenty of time to gather their thoughts and express themselves. Including some of TalkMoves and questioning prompts I discussed above would really compliment my CT’s already positive and interactive style in the classroom.

All our students can be math wizards when engaged in higher-level mathematical thinking!

Students are asked to put on their thinking caps when restating or re voicing their peer's responses and reasoning. WARNING: This may cause shock and astonishment when done for the first time!

Stepping into Teaching...A Classroom Worth Sharing

Inside the strong brick walls of this local school paints a simplistic picture of white hallways, old carpet, and a modest display of student work and, “Beginning of school,” displays. The hallways seem to go on forever, yet around each turn are classrooms full of new beginnings and tomorrow’s success. I see faces of diversity filling the seats while teachers, whose faces seem all too familiar to me, fill their minds with knowledge. The Resource room I am in this semester, is a classroom that separates students from their peers during reading, writing, and math, while also keeping the students connected, being just a few doors down from their general education classrooms. It is classroom that I see students excited to come to; and it is a classroom that focuses on instruction and learning rather than the flashiness of bulletin boards, themes, and posters. This pullout classroom has to be more. More than what the students can get in their general education rooms, and more instruction that can close the learning gap our students come in to these hallways with.

I have been in my internship going into the third week, and I can already see the struggles the students have in math. Math instruction is at the end of the day, and this reality in it’s own comes with elements the students must overcome (exhaustion, anxiousness, annoyance, etc.). This combined with innate difficulties towards mathematics makes for a long hour and fifteen minutes! The teacher tries to keep spirits up by reminding students of their number one job while in the classroom: to try! This message was written on the board after a pre-test that left two students in tears, a handful of empty test papers, and students giving up before they even get started on the lesson. This message has been up ever since, and it is a reference point that is used throughout all subjects and activities. I admire my teacher’s message, and will work hard to use it this semester in my lessons as well as in my own classes!

These, “Texas Math,” books are displayed proudly behind my teacher’s desk. They were intimidating at first sight, but after looking through the twenty plus books, supplemental materials, assessments, and games, I realized that this math curriculum being used offers many of the elements we are taught to include in our instruction of children with disabilities and differences. Differentiation is a big part of each lesson in this collection, as well as adaptations for students with language differences. My teacher has adapted the material itself, and only uses what works for her students. For example, after the negative experience most of the students had with the first pre-test given, she decided that she would no longer give a formal pre-test and only assess the students prior knowledge through informal observations during the lesson. Each lesson begins with a review of key concepts and skills that the students need, and she adjusts her instruction according to the students’ success or limitations during this time. This was an excellent example of making the curriculum work for a group of students. It was refreshing to see this first hand, and I am also looking forward to seeing how it continues to work throughout the semester.

In addition to the worksheets and activities presented in the curriculum being used, the students are given opportunities to explore mathematics using manipulatives. These are on an open shelf and are located in a kid-friendly area, which encourages the students to take advantage of them whenever they need. Inquiry-based activities are designed to use these manipulatives, and I enjoyed seeing students use them and include them in their strategies. From observing the students using these colorful cubes and squares, I also saw how their use prompted discussion as well as facilitate, “Ah Hah,” moments from the teacher and students. I would really like to see them used more consistently during the students’ warm-up activities, which I think would be a great opportunity for the students to explore and respond to word problems and other skills using ONLY the manipulatives. I think that would be getting the students to think on a different level than worksheets and other pencil activities do. I am wondering what other types of manipulatives can be used and applied to skills in math that are not computational and problem solving. I would be interested to explore those options more.

Vocabulary is very important to this Resource teacher! She does vocabulary warm-ups every day, which include a fun cheer or basketball dribbling of the words being used during the unit. I feel this is incredibly important for students with learning disabilities, as vocabulary can really strengthen your understanding of math. A student does not need to worry about knowing what the word problem or directions are asking, just how and why they are going to solve it. These vocabulary cards are included in the Texas Math curriculum she uses and when I saw them I fell in love with them and knew I had to post them! On the front is the vocabulary word in English and in Spanish. On the back is a picture describing the word and it’s definition. If it is read or taped for the student their use is appealing to multiple senses, and will help the student store the new knowledge faster! Not to mention, with the diversity present in our schools in Texas, the pairing of pictures with the definitions differentiates for our students in an effective way! I have to admit that the idea of a vocabulary focus in math is a fairly new idea for me, but I am seeing how important it is, and I am excited to be equipped with an easy teaching strategy to help our students succeed.

Small group instruction is a big part of my teacher’s resource room, and she utilizes this area of the classroom to provide scaffolding to students who are struggling. She identifies these students at the end of the lesson, and while the rest of the students move on to independent work, she pulls a few students to this table and provides additional instruction until she feels they are ready to be independent. This is an excellent example of flexibility and progress monitoring, both of which are critical elements of math instruction. With the use of paraprofessionals, and her trusty student intern, she is allowed this flexibility and can easily give students the attention they need. Within small group instruction the use of white boards are used to help the student demonstrate their work quickly, and provides the teacher with a fast and efficient way to check their work. It is easy for mistakes to be corrected, and they can make a quick visual organizer for the student to take back to their seats with them. The students love using white boards, and I like how it provides them a break from the standard paper and worksheets they are used to. White boards can be also be used during lectures for the students to copy down the teacher’s examples and follow along with. A place value lesson I observed really came alive using the white boards!

My teacher believes in celebrating the students’ victories, both big and small. She has a section on the wall where students can draw pictures of any type of victory they had recently and display it for everyone to celebrate. I really appreciate this vision because it not only makes the students acknowledge that victories come in all different shapes and sizes, but it also gives the students, who may not be used to having their work displayed on classroom walls, the opportunity to be proud of something for everyone to see. As I already told her, I plan on stealing this idea and applying it in whatever position I take on as a future educator.

Response to Reading Week 1

1. It gives the students the opportunity to use what they already know and how they learn well to solve the problem themselves. It also teaches them strategies (that they developed themselves) to use in the future.

2. I feel that my experiences with Math will make me a stronger and more effective teacher for my students. I can take the good and the bad experiences I had to facilitate an environment that will either encourage my students to have similar experiences or prevent them from having hard ones.

3. Time spent letting the kids discover and explore their knowledge will lead to more retention and require less repeated instruction in the future. While it is important to teach certain concepts explicitly, students will more likely retain basic facts and operations if learned in way that is meaningful to them (not being talked at). If we give our students the tools and strategies to continue to grow as a learner we are setting them up to be successful students and people.

4. It is not a good idea to ever tell a student something is, "easy." We also do not want to be too quick to jump in and give them too much information that prevents them from solving the problem on their own. A better way is ask them questions that will lead to their own discovery of the answer. These questions may help them organize their thoughts and give them the push they needed to solve the problem.

5. The article tasks focused on activating the students prior knowledge and utilizing their strengths (what they did know) to solve the word problems. The tasks also forced the students to make sense of the problems before solving them. In the marbles problem, the teacher guided the student through the problem (without providing hints and prompts) by asking him questions that responded in answers that allowed him to make sense of what the problem was asking him. He did the work with strategies and ideas he already knew and were comfortable with.

My Math Life Story

Peak Experience:

My math life story peak experience was actually born from a feeling of complete inadequacy I had in regards to my mathematical achievements. Due to my older sister's ability to excel in every math honors course she took in high school, I entered my high school journey with the already implanted expectation that I also would join the math honor's train. This expectancy however, did not last very long. So after two failed attempts across two years to test into a math "AP" class, I finally accepted my place in a basic Algebra I course during my junior year of high school. I decided that I no longer was going to try to prove to my parents and the guidance counselor that I was, "smart," enough to get that special credit on my report card, and I was going to just try to experience math, for me. Now, it may have been the teacher, the book, or even just the material I was learning, but boy did I increase not only my mathematical competence but also my confidence in my abilities to succeed in math. That year I went from a stressed out basket case trying to memorize every definition, equation and concept (for the test) to a math student who became engaged in class discussions, accepted help from tutoring, and eventually became a peer tutor herself. It is from that, "basic," Algebra I class that I took away more knowledge, and more experiences from which I have now transferred to my college math courses and assignments. I learned that sometimes you have to take a step back and forget about the formalities and pressures surrounding others' expectations. I then allowed myself to learn something for me! In the end it did not matter if my high school transcript said, "honors," courses or not. I moved on from high school with an improved outlook on what it means to be a successful student, on my own terms. It is this message that I will pass on to my students as a teacher. I will make sure they know that it does not matter where you end up on that honor roll, it is the journey you take to get there, and the knowledge you develop along the way. Teaching them to be the best THEY can be, and always remember, stress will not equate success!

Nadir Experience:

With the good comes the bad! As you just learned, I never did make it into a honor's math course in high school. My negative experience comes from that reality in combination with my school's choice to place a football/baseball coach in the position of teaching the, "regular," geometry class. His idea of teaching us was assigning chapters to read and practice problems for us to do inside and outside of class. This was done without any form of formal teaching. My frustration levels grew each class period, as I was unable to learn successfully by reading from the textbook. Eventually I began to shut down, and lose motivation to retain anything I was assigned to learn. My math experiences began to be about completion rather than mastery. Geometry was obviously not important to this teacher placed there to teach, so why would it mean anything more to me? Geometry became more of a social hour with the occasional rush to memorize angles and definitions in order to pass the test. That experience truly made me realize that lack of quality and meaningful instruction really does have negative impact on students. A large amount of important geometric concepts were not learned by myself due to this teacher's lack of ability to teach and motivate. I was forced, 4 years later, to play extreme catch up in order to succeed in a college level geometry course. As teachers, we should never set our students up for a deficit that large and influential. I will take my job as an educator very seriously as a result of this experience, and never take for granted the job I am assigned to do. I wish only that my students take away peak experiences in my classroom, and any nadir experiences that may come along will be quickly forgotten or celebrated.

Turning Point:

I do not feel that there is just one specific point or experience that sparked an important change in my math life. If there was one, my memory has failed me! However, a series of turning points in my mathematical journey came during my elementary math courses I was required to take before entering my PDS. The courses were not really about teaching (or re-teaching) math but exposing a different side of it from what we were originally taught; to understand the why. This was a turning point because up until then, similar to my colleagues, I thought math was full of very concrete, "do this to do that," type of concepts. When trying to explain why something is what it is, you have to take a step back and really dissect the process you took to get there. It is from these dissections that I realized that math can also be very abstract, and solutions can be influenced by our own learning styles and perspectives we take. While you can teach a child that two plus two will ALWAYS be four, you can also give a child multiple ways to solve addition, subtraction, multiplication, and division problems. They can choose which way suites their learning style. By choosing their own way to solve math problems they are more likely to experience success! I do not know about you, but I was only taught one way to set up a subtraction problem, and one way to show my work throughout my early math journey. These series of turning points, which came from my participation in these courses, revealed a more dynamic side to math that I can pass along to my students.

Other Important Scenes:

Around the World," is not a productive game! My third grade teacher loved to force us to play this torturous game after we all passed our multiplication times tests. It is from these games that my math anxieties stemmed. It is also from this game that the stereotype of female’s inferiority in math began to emerge, of course now I wonder if it was a more of a competitive characteristic that led to the boys’ dominance of being the last one standing. I still remember the butterflies I felt in the pit of my stomach when my turn to stand in front of all my peers and answer the random multiplication fact was next. I would consider myself an average to slightly above-average student in third grade. I can only imagine the anxiety my peers felt who were struggling learners when forced to stand up in the spotlight and display their memorization of the multiplication facts. These are feelings that I will work hard as an educator not to expose in my students. Those types of competitive games should be left on the playground!

My second scene comes a little later in my life. It was the first semester of my special education internship and I was placed in a fourth grade general education classroom. During this internship we taught a total of twenty lessons across the semester, dividing them up among math, reading, writing, science and social studies. For one of my math lessons I was told to teach the foundations for learning to divide. This lesson was taught over the span of two days, and by the second day my students were rolling along great and it was time to introduce them to the operational signs that are used to represent division. On the overhead, where everyone could see, I wrote a division problem using the traditional right parenthesis with vinculum, or this is what I thought. In fact I had actually wrote a square root symbol! I should have realized this sooner, but even after my students tried their best to expose this error to me, I continued with the lesson using a square root sign! It was not until my professor, who just happened to be observing me, told me after my lesson of my mistake. I was humiliated. I took the first opportunity to correct my error with the students and made sure they learned the correct sign that can be used when setting up their division problems. This memory just solidifies the critical idea that it is imperative that you plan, plan, plan and rehearse before teaching a room of children. Maybe have someone else look over my math lesson plans in the future as well!

Greatest Challenge:

The single greatest challenge I have faced along my math journey has been the time commitment I typically have to make in order to be successful in this subject. I feel throughout my life I always had to work just a bit harder at math than others, and no matter the problem, solution or expectation I always spent more time than my peers completing it. I have faced and overcome this challenge by dedicating more of time outside of school to understand math, as well as increased my engagement in discussions and math problems inside of school. I also sought the assistance of a math tutor outside of school during my high school years. This challenge never allowed me to slack off, because if I did I experienced failure (lack of accomplishment). With my math tutor, parents and sister, Allison, I gained the support and help I needed to be successful and accept this challenge with full embrace. This challenge has impacted my experience with math in the form of confidence in my ability. I have come to accept also that maybe it was my lack of confidence all along that created this challenge for me. When my confidence was low, the time commitment increased, but when my confidence was a little higher, the time commitment decreased. This mutual relationship has remained consistent throughout my childhood and on in to my adulthood.

Special Education Teacher:

I want to be a special education teacher because I believe that every child can. Not only do I feel every child can, but also that every child should be given equal opportunity to reach their inner potential and exceed expectations. One of my favorite quotes from a Christian song I love reads, “Impossible is not word, it is just a reason people give for not trying.” Throughout history that “word,” "impossible," has been used too much when speaking about and teaching children and adults with special needs. It is amazing the change that has occurred in this field and I want so much to be a part of that. No matter the disability, I feel that children are all born with special gifts. It is with these gifts that we as teachers can really make a difference in their lives, and it with these gifts that they can have meaningful growth. I want to celebrate the, “special,” in special education!

When it comes to teaching mathematics to my students I feel that their success will have to start within me. I need to be confident in what I teach them so that they in turn are confident in their ability. I believe that there is no one way to teach or learn. I will be sure that they are given the tools to solve the problem the way they can be most successful with. My vision of math instruction also includes cooperative and collaborative learning. Breaks in the lesson when my students can explore and learn together with their peers. I will do all I can to prevent math anxiety from occurring along their math life journey!