On Composing Meaning

In the journal article, Constructing Meaning: Standards for Mathematical Practice, the writers discuss the differences between modeling with mathematics versus modeling mathematics.

Modeling mathematics "is using concrete materials or other visual representations to clarify or give meaning to mathematics" and modeling with mathematics is using mathematics to describe and/or explain a real-world context.

The teachers that were involved in the summer institute that the research was based on did certain standards for mathematical practice and reflected on their day's experience.

This distinction makes sense, because one term discusses using manipulatives to explain a mathematical problem and the other is to use math to describe a situation.

The teachers worked through a problem and then reflected on what they did. It is important to provide teachers with this distinctions because it provides insight in the language of teaching.

Open Curriculum. . . Open Mind

Open curriculum is an idea I am certainly implementing in my own classroom. I love the idea of having a teachers manual and resources as a back bone of what and how to teach certain subject matter. However, often I venture out and use outside teaching strategies and resources to complement my approach. The article really does a nice job encouraging teachers to venture out as not all set of students learn the same way as perhaps the book anticipates.

Do any of you feel the same way? Is your school supportive of an open curriculum approach?

Scripted math programs don't know our students

Scripted math programs don't know our students.

So how can we have any lesson plan script in the math classroom?

Here are the many ways scripted lessons are failing our students:

They're not leveled. Every classroom in America has students of all different levels. A scripted math lesson plan cannot possibly be teaching ALL the students in the classroom. If you believe in student-centered learning -- which I do -- you can't possible use a teacher-centered/scripted lesson. You must meet the individual student where they're at in their learning.

Scripted lesson plans are on a strict time constraints for each section, so you miss all the exploratory elements of true math inquiry -- whole group, talk it out, think it out lesson.

Scripted lesson plans typically offer only one preferred method or style of answering a math question. There isn't any variation in problem solving, and if there is it's limited.

What's the point of the teacher if the lessons are scripted? Why not just have a robot in front of the classroom, reading each portion of the lesson, no skill, no training, just a great facilitator of scripted lessons? A scripted lesson misses the human element -- the teachable moments -- that are always more important when the lesson is taking place than the nuts and bolts of the lesson.

So its great to "open spaces" in curriculum but why not toss these scripts all together and just jump in.

Opening Curriculum

I found this article to be quite insightful into the way teachers are supposed to teach math versus how teachers want to teach math. I’ve said before that I struggle with using our math curriculum and would much rather supplement my own lessons. This article follows the idea that students need to develop more skills and knowledge than current math curriculum allows. The article discusses different strategies to help build students’ MMKB (multiple mathematics knowledge base). The first strategy aims to manipulate the sequence of the lesson, or even cut out parts altogether. Rather than always following the same pattern of instruction (opening, lesson, practice problems, word problem), change the order and allow students to problem-solve first. The second strategy invites teachers to differentiate and adapt practice problems so each student can be successful. Teachers should encourage students to complete practice problems at their individual level using their own strategies and methods. The final strategy discussed is to make math relevant to the students. Build connections between math and real-world problems.  Using these strategies may not solve the problems that we are having with teaching math, but they definitely may help students stay more engaged and motivated to learn. 

Dig Deep for More Effective Lessons!

Amazing Rainforests, Jesse's Train, and The Geometry of Scoliosis are three new lessons to add to my stock that explore elementary mathematics.  The lessons are inquiry-based, allowing students to explore the manipulatives and/or information and use questioning to get to their answers.  They work in pairs or groups, documenting their progress throughout.  Then, they share out their findings whole group and discuss the strategies each group used, critiquing their findings.

I really love these lessons especially since they are cross-curricular, integrating science, social science, and language arts.  I’m  always looking for ways to make math time more exciting; special.  I’m going to try Amazing Rainforests soon!  Which one of the three lessons did you find most applicable to your math instruction?

Number Sense Intervention

After reading the article about a Kindergarten student named Kyle and his struggle with number sense, it became clear to me that there are so many students struggling with the exact same problem in schools today. The NSI system made so much sense to be and seemed like it would be very successful for students. I really liked how it emphasized beginning to work with numbers 0-3 and establish a firm understanding of those numbers before moving on. I also thought the chart was great that showed each number being one more than the last and one less than the next. This is a simple, yet so important concept that students must achieve in order to be successful in math. I agree with the language they used as well, such as, "one and one more is two, two and one more is three, three and one more is four, etc." This really shows and explains to students that each number is one more while also introducing addition. They understand the larger the number the larger the quantity. This concept may seem simple and quick to teach, however it must be thoroughly taught and looked at through different ways. Understanding a strong sense of numbers is crucial to further number calculations, word problems, real life problems, mental math, the list goes on and on.
NSI is a great system for struggling students and should really be further understood by all teachers teaching in the primary grades!

Math as a Social Activity (late post)

A teacher's ROLE...

I found the article from Perspectives of Early Childhood Education very interesting. It gave much insight as to what teachers are doing now and the role that we play on developing our students' powerful minds. One part of the article I liked was when it talked about assessing a child's performance; teachers need to use a variety of approaches to access learning so that children feel comfortable about evaluation as a natural part of learning. I find this very important because I am always doing informal assessments with my students. Many of my students (and myself) see the word QUIZ and freak out! I try to make my assessments natural so that the students do not feel overwhelmed and are able to perform to their best ability. This article also states that it is NCTM's vision that by 4th grade, students should be confident in their abilities to do math. This has to be done by making sure a strong foundation is built and that we bring in real life situations for our students to understand the bigger picture of how math relates to their lives. Awww, the pressure is on. =)

Formative Assessments

As teachers we know how important it is to allow assessment to drive instruction. For the most part, we are assessing our student’s progress all day. This article proposes the use of a “palette” of formative assessments for teachers to use. The assessments include observations, interviews, show me activities, hinge questions, and exit tasks. I definitely adapt lessons through my observations in the classroom and use activities to further assess understanding. The only element of this palette that is new to me is using “hinge” questions. These are strategic questions asked at “hinge” points in a lesson. These strategic questions allow teachers to quickly assess student understanding and guide instruction for the following day. Perhaps it is necessary to reteach a concept or perhaps it is okay to move on to the next skill/lesson. This is another tool that allows me to monitor my students’ progress and adapt my instruction to meet their needs. 

The Integration of Real Life Concepts and Math

I have always been a great supporter of integrating multiple subjects in to a lesson and bringing it to life by incorporating real life situations. For me as a learner, this was the best and most effective way to learn any skill or concept because it allowed me to manipulate content matter in a way that was applicable in real life. It was easier for me to make sense of this situation and understand when, how, and why to apply a concept. I also found this to be true for my students. The students in my classroom seem to show more enthusiasm and engagement when content matter is associated to something they love or show interest in. Similar to Amazing Rainforest article where students in grades K-6 explore math concepts, which are integrated with science and social studies content matter.  In the article students learned about math skills such as the number system and place value by using the rainforest as their setting for exploration. The Unit outlined knowledge building upon each lesson while also integrating aspects of science and social studies. This is something I will certainly love to integrate in to my unit of number system and place value with my wonderful 1^st graders.   

Reflection on Count On It-Congruency Matters

When you are an accomplished adult, it is difficult to remember how it was when you were acquiring elementary skills such as holding a pencil.using scissors, counting, and one-to-one correspondence. When one becomes a teacher (or a parent) and one is teaching these skills to a child, one sees that these skills, though fundamental, are sometimes a challenge to teach. The journal article Count On It- Congruent Manipulative Displays explains in detail many ways in which manipulative displays could help a student connect concrete mathematical ideas into semi-concrete ideas.

One aspect that is challenging when using manipulatives is management. How do you introduce manipulatives to children who still think of these objects as toys? Also, when you do use manipulatives, are you using a manipulative that will connect the ideas you are teaching to what the student is experiencing. Morin and Samuelson go through important descriptions on how, when, and why to use manipulatives. One example that they describe is how to work with the student who sees manipulatives as toys. The other example that I found helpful was that if the students are counting apples and oranges, to use printouts of apples and oranges as manipulatives, until the student is able to connect the number to less concrete representations such as tally-marks.

I found the article enlightening and helpful. I had not thought to use manipulatives like printed and cut out apples and oranges. This step is necessary for some children until they are ready to use other counters. I also found the discussion of how to manage/ teach students how to use manipulatives so that the use of manipulatives is not destructive and messy.

Congruency Matters

I read the article titled, “Count on it- Congruent Manipulative Displays” by Joe Morin and Vicki Samelson because I am such an advocate for the use of manipulatives both in and out of the classroom.  The National Council of Teachers of Mathematics encourages the use of manipulatives and recognizes how most children flourish from the interaction but if the connection and concept aren’t made clear, then confusion can result.  I am aware of the many benefits of incorporating manipulatives, especially for the purpose of teaching new mathematical concepts and showing students new ways to comprehend and view existing knowledge.  I also think manipulatives are key with English Language Learners! I am vaguely unaware of the detriments of using manipulatives because I have never had a student not somewhat grasp the concept.

Has anyone experienced any significant problems when using manipulatives to facilitate and foster a students understanding and growth with a certain concept? 

Distraction is a caution to be aware of but more importantly, improperly displayed manipulatives can threaten the conceptual continuity of the materials. For students to use manipulatives productively, they must be able to make a symbolic connection as a quantity (blocks) and an individual connection (many blocks can build a tower). Conceptual congruence is also very important for teachers to be aware of because if you are asking students to compare two groups and the manipulatives in each group are different in size or shape, then visually the student lacks clarity and may focus his or her attention on the objects themselves rather than on the attribute of cardinality.  Congruency between numerical concepts and manipulative displays is highly important for all students and if not properly introduced, it can lower the level of their potential!  If congruency exists then it will help facilitate student’s transition from concrete to formal abstract mathematical knowledge and help them relate to real world problems! :)

NSI

NSI (number sense intervention) makes an enormous amount of sense to me. I have been very vocal about my dislike for our current math curriculum (Envision). I don’t like the scope and sequence of the program, nor do I like that the curriculum only allows for a one way strategy to solve problems. Our current math curriculum concerning number sense goes through numbers 0-10 focusing on one number a day. However, there is no building on number relations for students to connect numbers and see patterns; instead the numbers are taught exclusively and separately. The NSI uses a visual/spatial technique to help students learn to count and develop connections between numbers and counting. It also serves to help students understand parts of a whole, as we introduce addition and subtraction. I think it is a great idea to be nurturing number relationships while students are learning to count and learning the basics about numbers. Why separate the two when they go hand in hand?

"Borrowing" VS "Regrouping"

After watching Liping Ma video about regrouping and reading a bit of her book, Knowing and Teaching Mathematics," it is clear that she really knows great methods of teaching math. She displays a chart in the video that shows American teachers vs Chinese teachers when it comes to teaching "regrouping" or "borrowing" methods. It shows that in American, teachers tend to use the vocabulary  "borrowing" when teaching subtraction of 2 or more digit numbers. In China, teachers primarily use the word "regrouping" when teaching the same concept. She states that in China teachers emphasize that students are simply "renaming" the number, not borrowing. She further shows that American teachers typically teach one way of doing subtraction whereas teachers in China show many ways of teaching this concept.

Of course we hear all the time to provide multiple ways of teaching a concept. In yet statistics show that American teachers as a whole as just not doing it. This video and reading has definitely opened my eyes to different ways we can teach this one concept in our classroom to make it easiest for the students to learn. For example, Liping Ma demonstrated how to regroup numbers into smaller numbers that will be easy for the students to subtract. Such as regrouping the number 53 into 40, 10, and 3. Smaller, whole numbers that students can easily work with.

It all makes sense to teachers I'm sure, but maybe they find there is not time or wiggle room in the curriculum for teaching all these different methods. I feel that it is a necessity that students are provided multiple learning tools for solving a problem. It is also very interesting to me that a simply vocabulary word change can make such a difference in student's learning. What do you typically use when/if you teach this concept in your classrooms? Borrowing or regrouping?

A Reflection of the Transformation: Slater School

The article Transforming Teacher Learning to Student Learning-Slater School, reminded me of the school I currently work at and our transformation as a staff. We are currently undergoing an internal transformation of how to best and effectively teach our current school population. As a staff, we are learning how to teach our students best by applying “The Core Instructional Model,” which is our districts way of preparing us for success with the common core state standards. The model highlights academic learning time as student driven lessons which gives students access to manipulate content matter. The lesson is guided by student friendly objectives which allow students to monitor and self asses their progress. 

Although as a staff we all share the same goal for our students; we also all had different means and conceptual understandings of how to go about it, similar to the teachers at Slater School. Their was a shared feeling of confusion at the start of our transformational program as well. Some teachers were very reluctant to implement any change despite efforts of the district.  

Nevertheless, the process shared by the article made me truly reflect on our experience both as a staff and individually. The article presented many similarities in the overall process and learning expectation. 

Reflecting Rocks!

In reading, "Elementary Mathematics Pre-Service Teachers’ Learning from Cases," Henningsen studied pre-service teachers and determined that case discussions are powerful tools for engaging teachers in reflection.  She found that seventy-five percent of the teachers she studied said they learned that students should be explaining to each other and debating with each other and that such activity promotes mathematical learning.  As a teacher who has been working hard to make time for explanations and discussions in the classroom, I can say that half way through the year, it really started to pay off and I felt my students were far more comfortable with the process of debating and explaining, not only when they felt their answer or the other student's answer is correct, but explaining what went wrong.  Students are paying much more attention to precision and accuracy when they analyze their work and the work of others.  They learn new perspectives and new strategies and they learn the invaluable skill of collaboration.

Additionally, she found that 50% of the participants said that the cases opened their eyes to the fact that teachers "should not underestimate students’ abilities to explore mathematical concepts and make sense of them on their own."  It's true. I am amazed at the exploration that goes on in my room.  Working in different heterogeneous and homogeneous grouping patterns makes a big difference in their acquisition of the material.  17 out of the 20 participants stated that grouping students worked as a tool or means to promote communication. In my EL kindergarten classroom, I find that making time for explorative communication activities improves my practice and pushes my students to practice standard English regularly.  The pre-service teachers clearly benefitted from the reflective practices with more than one-third stating that they learned new insights about mathematics.  How do you reflect on your practice and how valuable do feel classroom discussions/debates and group work are?

Sometimes Teachers Need Coaching Too....

After reading the article, "Classroom Coaches Critical as Teachers Shift to Common Core," I have a better understanding for the need of learning coaches in schools. With the new Common Core standards it has become apparent more than ever the need for learning coaches. This article focused on math coaches in schools and describe the success of the students with a math expert also present, in addition to their teacher. It is great for teachers to get an expert's advice on teaching math and maybe incorporating new materials into the curriculum.

The article mentioned coaches that were trained and had expertise in the common core standards. I think this would be very beneficial for teachers to have in the classroom because teachers are so busy making lessons, communicating with parents, planning, meeting the needs of their learners, etc. Teachers are just SO busy! I can imagine how helpful it would be to have someone helping to make sure lessons are meeting the needs of the common core standards and ensuring that each standard is met. I'm not currently a teacher right now, but I could definitely see this being beneficial for teachers today. What do you think?

Digging Deeper- challenges teachers face

I feel like administrators often blame teachers and the teachers often blame the administration for the problems that occur in elementary schools today. Why are we always blaming each other? It is so frustrating to me that the expectations of students are different at every school. Some administrations do not care about the individual success of the students, but rather the overall test scores. Other administrations acknowledge each students' academic achievement. It is hard for teachers to not only have to do what is best for the students, but please their administrative team. Many administrators have unrealistic expectations of teachers. Some teachers are very lucky to have a supportive administrative team, while other teachers do not. Teachers need support from administration!

OHHH, I get it now.

I looked at two of the digging deeper articles. The first was an article discussing reform, revolution and paradigm shifts in math education. This article was interesting because it talked about the way content is being taught across all grades. The study focused on pre-service teachers preparing to graduate. It addressed many things that I found interesting. One aspect of the article that I liked was the fact that the pre-service teachers explicitly stated that the cases they looked at helped them to understand how much a teacher is obligated to observe student thinking by continuing to ask questions and get clarification from the students.

I feel that this idea is a re-occurring topic across this course; the importance of making sure each student is able to clearly explain their mathematical reasoning. It is so important as teachers to take the time and ensure that all our students can communicate the objective of the math lessons being taught.

Another great aspect of the article, was the last few sentences that state the importance of a teacher to be able to look at students work through their eyes and be able to view multiple perspectives. Which is something that we all know, but it is a good reminder.

The other article that I looked at was by Liping Ma. It addressed similar ideas about allowing students to express "flexible thinking" and it also talked about making sure that students don't become "locked in" to only one approach to solving math problems.

Again, these are all great ideas! I am teaching long division right now and can see how many of my students can get confused over the topic. I have showed them multiple ways to solve their problems and I have also showed how the problems are broken apart. When I was doing this last week, one of my students said, "OHHHHH, I get it now!" That's probably one of the best phrases you can hear as a teacher.

Laying the Foundation: Quality VS Quantity

As elementary school teachers, we are responsible for laying down the foundation for student achievement. We are the blueprint to their future, therefore, it is imperative that we are not just teaching to the test or just to say that we covered a concept. But, teaching in a way that ensures that students are comprehending and taking the information being extended and plugging it into their long term memory. I feel passionately about this about this because once students get to middle school the classroom  sizes dramatically increase, there is less one on one time with teachers, and students are forced to become more independent. Also, students who feel inadequate, are more likely to shut down. Once a student shuts down, it is hard to re-inspire them. Again, it is quality, not quantity.

The Life of a Teacher...

The Challenge of Implementing the Common Core State Standards for Mathematics by Michigan State University once gain shows some of the handicaps with the public school system. It is unfair to blame teachers for things that are out of their control such as lack of materials and text books for teachers. But, I do feel that teachers should have high expectations being we are expected to meet standards and ensure our students meet their full potential. With that said, we live in a society where there are resources everywhere and if a school is unable to provide the necessary materials, as teachers, we should be proactive and get them ourselves. This is something that I do at my school regularly and it has actually worked to my benefit for it has made me more creative, resourceful, and strengthened my researching skills. 

I an convinced that the decision makers (administration) rarely has the students best interest at heart; therefore, don’t expect much from them. 

On ‘Elite PD Programs Seeks to Build Top Teachers’

I read ‘Elite PD Programs Seeks to Build Top Teachers’ and I am disturbed by the price tag of $12,000. My first question is “Who is paying for this?  “Why invest money in an organization that is not transferable or trusted? The second thing is that this program is not connected to any university but it is a for-profit organization.

I believe in education and continued learning. I distrust “for profit” organizations and business models in regards to schools. I believe in lifelong-learning. Who believes that business people know anything about the process of teaching and learning? I know business people want to make money by separating me, school districts, and other fools from their money. 

I simply do not trust anything that a businessman knows anything about child development, pedagogy, language acquisition, etc.

Business people might know how to make connections with the community. In my mind, the business person’s interest is the bottom line, which is profit, in taxpayer's dollars.

Digging Deeper Class 5 Articles and Videos

In the Digging Deeper Class 5 Articles and Videos, there were three articles that ere related: Procedural Fluency, Making Ten, and Undoing Rote Memorization.

In the article, people were both pro and con for either methodology. I think because of the Common Core, and the public’s unfamiliarity with pedagogy, we are seeing extremes in the polarity of opinion in support of the traditional (rote memorization) and the popular (procedural fluency, such as making ten).

I am in both camps. I think that we need both. I believe that it is important to teach procedures so that when, for example, we encounter an equation (algorithm) we know how to solve the problem economically. To do that, we need to know which procedure to use and why. That is Procedural Fluency. Speed is also important. That is where rote memorization comes in.

Writing also demonstrates the need for procedural fluency and memorization. To write correct sentences, a student needs to know grammar. Grammar is taught explicitly, and it takes time to learn the rules to write and speak English correctly. Spelling is also needed to write correctly. Spelling is usually learned by rote memorization, sorting, and noticing patterns in word formation.  Could you teach writing without focusing on these integral parts of language?  Maybe.  However, I think students benefit by explicit instruction (procedural fluency) and practice (rote memorization).

If the public understood this, maybe there would be less polarity of opinion in the need for both procedural fluency and memorization.

Digging Deeper: Using Video Based Pedagogy in Elementary Math

As person who is curious about so many topics, I have come to rely on the internet to learn about many subjects. It is no surprise that many instructors are sharing what they have learned about teaching in the internet. 

When I see a video on pedagogy, I try to see the video through different lenses. That means that I see it as a student (Do I understand the concept? Do I know what the instructor is talking about?). I also see it as a teacher, (Have I ever taught the subject in the video the same way?) (Have I made the same mistakes that the instructor is making?).

When I saw the video Math with Manipulatives - Subtraction Using Base Ten Blocks, I saw the video with critical eyes. I have used a document camera to teach subtraction, using base ten blocks. I think I have taught the same way as the instructor in the video. But as an observer, I found the constant moving back and forth from the writing confusing.

I think overall, the instructor was good but next time he should be able to write without taking the card out of the field of vision. I need to apply that in my teaching. I am deeply grateful for the tools we have (document cameras and manipulatives), but I am aware that teaching comes down to transferring information with the least amount of confusion. What I learned is that as an observer, even minor disruptions can create confusion.

The second video I saw was an informative video on Singapore math, which finally clarified what Singapore math is.

The third video I watched was How to Teach Elementary Math Through Real World Data | eHow. It is interesting to hear someone teach math using Real World Data, think it is too elevated for your grade, and then realize that you are using that same information in class.

Lastly, I am thankful that videos on pedagogy exist and that they are available for anyone to see. These videos are helpful in so many ways; 1. To teach a concept to a teacher, 2. To provide ways to present the information, so that we don’t reinvent the wheel, 3. To help us reflect on our own methods of communicating these ideas and be more clear. 

Our Challenge as Teachers. . . .

“The Challenge of Implementing the Common Core State Standards for Mathematics,” presented by Michigan State University was fairly accurate in my eyes. The document presented disappointing facts however, it fell within the margin of where most schools currently stand. The unfortunate percentage involving the lack of parental support, lack of textbooks and other resources is a large obstacle to implementing the more challenging CCSSM. It amazes me how some district set such high teacher expectations without facilitating the necessary resources required to meet those expectations. I believe this is in part the reason why there is such a high teacher turn out rate the first few years of their teaching experience. It can be overwhelming for some especially when you don’t have sufficient support.

 This will be a never ending struggle in education. The constant battle of accountability in the classroom, within a district, and parental involvement will remain a constant shift. It is unfortunate, however a unfortunate reality. Will we as teachers ever have “enough” or all necessary resources to meet each students needs?