Prior to reading any articles, I typically read titles, subtitles, and skim for basic information.  This technique is useful for any reader.  Of course the anxiety potential is heightened when seeing words such as polynomial, linear equation, and factoring trinomial.  My thoughts become twisted and stomach begins to knot up.  Immediately, thoughts go toward giving up and throwing the articles in the trash. There will be no reflection on this week.  None the less, I trudged on determined to connect and relate to the articles in some way reading again and again until there is some way found to connect and relate mathematics to language instruction and literacy.  Eventually a relation, a connection is realized regarding using concrete models to introduce concepts, increase and learn important vocabulary, and using alternative approaches to provide students access to mathematics understanding

Annette Leitze and Nancy Kitt believe learning algebra is possible for all persons by using concrete models such as the algebra tiles mentioned in the reading.  The authors suggest this technique gives students a series of levels to learning: concrete, pictorial and abstract or symbolic. This is true for all subjects and pertains to not only mathematics instruction but language based courses also.  Young children especially gain experience going through various levels and stages of learning beginning with concrete and progressing to the abstract.  When working with students who have various learning styles and special education services, concrete is a great first stage of presentation.  For example when working with a deaf student on vocabulary building, I can say the word cup, sign the word cup, show the printed letters for the word cup, but unless I show a picture of the cup or an actual cup the concept of cup may not be acquired or fully established.  Although a child may have been shown how to use the cup, without being able to relate the cup to everything, pictures, sign, print and tactile – full understanding of the concept or meaning is not established.  It would be interesting to do a study and collect data with deaf students implementing all steps or methods of language or vocabulary acquisition.

Next, Rhela N Rubenstein focuses on strategies for increasing and learning mathematics vocabulary.  Students with limited vocabulary as a result of language delays may easily become confused during instruction, for example when math words such as sum and some are used during instruction.  A deaf student relying on speech reading and lip reading skills may have difficulty understanding which word is being used if enough clarification is not provided.  Although students using sign language will understand the differences in meaning, students using speech reading and listening skills may become confused or misunderstand these two words since they both look exactly the same on the lips.  This article in particular states the importance and the need for early language and vocabulary work.  It is important for parents and teachers to show, explain, describe, print, and give details and concepts as early as possible with students having language delays.  This suggestion does not come from knowledge gained through research, but as a parent these are issues that popped up from time to time when my daughter was young. Children born with hearing loss or deafness are at an instant disadvantage. Without sound may of the ways early language is acquired such as environment, parents, radios, television, and other siblings are not available.  Language stimulation is acquired from one on one and face to face interactions.  This is but one challenge children with hearing loss face in the classroom.

Another challenge in language acquisition is having access to skilled interpreters.  Interpreters do not always have a good grasp on conceptual signing. Using the same sign for two words with different meanings such as some and sum could seriously impair a child’s understanding of mathematical concepts.  This unfortunately occurs more often than I care to discuss. Additionally, these authors asked “how do we help students recognize these nuanced differences of usage?”   The idea of the need for understanding the difference in meaning is enormous.  Students using speech reading and or sign language need to be exposed to all meanings.  Signs must be conceptually correct; Interpreters must have an understanding of mathematical concepts and meaning as well as parents.  These concepts must be expressed, signed, spoken, practices, and shown to ensure the correct meaning is acquired.

Last, John W. Coburn discusses using various approaches to teaching algebra.  This article seems similar to several of the earlier articles read this semester regarding students invented algorithms and teachers teaching from the same, old approach.  Although, I have solved inequalities, graphed quadratic equations and factored polynomials, my heart races and anxiety increases while reading this article.  I look at some of the text and feel as if I am reading a foreign language.   There are pieces of my mind wanting to shut down and give up.  Equally, the intrinsically motivated side of my mind is determined to give this another chance, to reread over and over until something stands out.   When Coburn discusses students using certain uncommon styles, I am intrigued and continue reading.  I absolutely agree with this discussion.  There is a lot of information which seems to support providing students multiple ways for solving mathematical equations.  It is essential  to use every available resource in an education setting and these resources should be used until the concept is acquired.

I am beginning to understand there is much more to being math literate than merely knowing math related vocabulary words.  When hearing the term, “math literate”  the association to this term has been to the meaning of words and not concepts.  After reading and studying ideas pertaining to math literacy, it appears becoming math literate follows a process similar to learning to read.  Students must acquire and understand specific components necessary when moving from the learning to read stage to the reading to learn stage.   Thus, if students begin to make decisions about being math smart or math dumb during early middle school years, classroom environments must be established allowing teachers to implement strategies and practices needed to build student confidence and knowledge of mathematics stages and provide opportunity to promote and excite students about doing math through effective discussions, good classroom management and building on student geometric concepts.

Smith et al., (2009) presents the Five Practices Model to promote student participation and effective discussions during mathematics instruction which assist with offering a more manageable group time.   Planning a lesson to the last detail including student anticipated responses and solutions is a key component in being able to offer the best mathematics instruction while reaching the most students.  According to Smith et al, having a variety of scenarios established and knowing which direction to focus student discussion based on purpose of the lesson makes wise use of instruction time.  I, as the teacher, must be able to think about and offer multiple solutions to problems and explanations of concepts students might come up with.  This creates some anxiety as I ponder my ability to understand multiple processes and be able to think about all the ways a student might consider solving a problem both correctly and incorrectly.  In order to adequately facilitate my students, I would consider it essential to collaborate and discuss lessons being planned with a more experiences math instructor.  I agree with and support the use of a constructivist approach in teaching mathematics.  A constructivist approach, I know is successful in reading instruction and feel this would be a great way to present math instructions.

Although this model is a great avenue for conducting student centered, whole-group discussions and instruction, I must be able to effectively manage my classroom and instruction time.  Students need to be able to talk and write about math.  Yet, as steps are being built to understand mathematical concepts it is equally important to build steps for appropriate discussion of math techniques.  Group discussion can easily move off task when a teacher is unable to effectively manage the class.  For instance, last week we talked about using hands on work and activities, and how without good classroom management these tasks can easily become chaotic.  The whole purpose of the lesson could be lost without a focused instruction plan and good management.  It is important to anticipate the direction student discussion may go and how to bring it back to the main focus should other ideas become the topic of discussion.   When discussions move off topic, it can use valuable instruction time and trying to get back to the main point becomes difficult.  For me, one way to keep instruction on point is to know specific levels and needs of all students in my classroom.  This knowledge can help with planning for possible scenarios and having responses and instruction geared toward keeping everyone on the lesson task.

Similar to reading concepts, math geometric concepts are built on in layers.  Middle school is a good time to build on students thoughts about geometric shapes and special sense.  Van de Walle suggest,  it is important as a teacher to be aware in grades 5-8 there will be students at varying levels and in order to move to the next level a student must be able to understand each layer presented in the van Hiele Theory of Geometric Thought.  Each level, content area, overlaps and builds on each other.  This seems very similar to the components needed to learn to read.   Considering the importance of parental involvement in student awareness of print and reading components in the early years, I think this same idea must apply to students learning math concepts.  Components necessary to build on one’s ability to understand and then apply knowledge to different concepts may not be focused on at home in the early years leading to difficulty obtaining basic information necessary when entering elementary and middle school math classes.   Yet, this is an area parents can work to provide developmentally appropriate practice for the student. I believe education and language level of parents do not necessarily impede learning of basic math sense.  Many parents recognize shapes and can talk about these in a variety of ways building on those lower levels of geometric sense which may have some benefit in the classroom.

Parents should be encouraged to provide early literacy opportunities in all subject areas including mathematics. I work to provide resources and support in reading literacy and focus on parent involvement in helping students to improve reading and writing ability.  However, it is becoming more apparent the need to work with families to provide support for their children in mathematics. Both articles present models which with some changes implemented to fit with the parent perspective might be useful when teaching how to work with and benefit students outside the classroom.

 Furthermore education has placed such a focus on reading and writing, I am concerned early learning moments for math concepts such as spatial sense are left out of the day’s instruction.  Also, more and more art programs are being cut from education budgets.  Early drawing in art classes gives students a sense of shape and space in a non threatening learning environment most children enjoy giving the child and opportunity for spatial sense and development of geometric thought an opportunity to grow. As education seeks to work toward improving the mathematics literacy in the United State, it would be interesting to study the implications of removing art programs because of budget restraints.