home

=Six Principles for Teaching Mathematics =

The six principles for teaching mathematics refer to the principles identified in the Principles and Standards for School Mathematics published in 2000. Even though these principles and standards are not enforced nationally they do guide the laws of the most state policies in the United States.

Purpose of this Wiki Page
This wikispace is dedicated to improving mathematics teaching for beginning teachers. It is the goal of the editor that beginning teachers will learn how to take responsibility for student learning through assessing their ability to effectively apply the six principles of teaching mathematics developed by the National Council of Teachers of Mathematics (NCTM). Beginning teachers will explain the six principles of teaching mathematics and identify dispositions and traits of good mathematics teaching that they can develop and teaching practices and assess with respects to student learning.

The editor realizes that a teacher's beliefs related the NCTM's principles of teaching are formed from their past experiences. Therefore, on this page beginning teachers will express both teaching questions and practices they feel are related to each of the principles. The editor's purpose in publishing the beliefs of beginning teachers is to motivate discourse on improving mathematics teaching.

=Six Principles=

**__Overview:__**
“Excellence in mathematics education rests on equity—high expectations, respect, understanding, and strong support for all students.” (NCTM)

All teachers are responsible for providing an equitable education to each of their students. In order to do this a teacher must address the individual needs of students and integrate ways to support them. By creating an equitable learning environment the teacher maximizes the potential for each student to learn.

Equity is often confused with equality because in many scenarios they serve similar purposes. However, in education providing an equal education would be mean to give every student the exact same thing. Since students are made up of diverse learning styles, backgrounds, and abilities, they may need different types of support in order to succeed.

Assessment can contribute to students' opportunities to learn important mathematics only if they reflect, and are reinforced by, high expectations for every student. -- Mathematical Sciences Education Board (1993, p. 109)

Different cultural groups in U.S.society may have different intellectual traditions that create different conceptions that create different conceptions of reality than that tapped by our testing instruments. -- George F. Madaus (1994, p. 80)

**__Resources:__**
[] [|http://www.as.wvuf. ".edu/~equity/equity.html] Madaus, George "A Technological and Historical Consideration of Equity Issues Associated with Proposal to Change the National's Testing Policy." Harvard Educational Review 64 (1994): 76-95 Mathematical Science Education Board, National Research Council. Everybody Counts:A Report to the Nation on the Future of Mathematics Assessment. Washington DC: National Academy Press, 1989

Questions teachers should ask themselves about equitable teaching:
__How is the role of students' backgrounds and experiences recognized in judging their responses to assessments?__ //Assessment practices must promote equity by a concern for all students.// __How can a secondary teacher, with only 50 minutes of class time, ensure all students are meeting the standards and getting the resources they need for success?__ //This is important because secondary teachers only have a limited time each day to teach their students. The curriculum is usually set and very rigorous. Teachers need to know how to reach every student within this limited time frame.// __As mathematics teachers, should we strive to change curriculum to make it culturally responsive?__ //This is important because in order to teach equitably, we must meet the students where they are at. We can not expect students from different cultures to learn in the same ways-it is necessary to use equitable teaching styles and diverse content (while still meeting content standards).// __//What are the gender bias issues with student/teacher interactions and what can be done to limit possible issues? //__ //Research has shown that teachers have had a tendency to favor one sex over the other in the classroom. Most teachers seem to be unaware of the fact when they are questioned. The teacher’s sex does not see to be a factor in determining the outcome of the bias. It is the sex of the student which is the determining factor. Studies have shown that more males receive the greater amount of attention (acceptance, praise, criticism, and remediation). The tendency is for science and math classes to be biased more towards males than females. Males are more likely to receive assignments to higher ability groups in mathematics. Differences in the treatment of girls have led to girl’s lower self-esteem, lower self-confidence and reduced risk taking. Expect high expectations from everyone to remove stereotype and gender bias issues.// __Why should we, as teachers of mathematics, ensure our classrooms are equitable? Does being a math teacher make you more responsible for making your classroom equitable compared with teachers of other academic disciplines? __ //This question is important because of the gender and race biases in a mathematics classroom. Because of the competitive nature of most math classrooms, female students may have difficulty learning or enjoying being in the classroom. In addition, there are race biases--for example, don't assume that an Asian-American child is good at math simply because he/she is Asian-American. To make your classroom equitable, you must have the same expectations for everyone.// __How can I systematically assess myself on how equitably I am teaching my students?__ //This question is important because it will be very easy for me as a teacher to be biased in my teaching (and thus inequitable) without ever being conscious of my biases. One way to address such potential biases is to interact with students in a more regimented and systematic way, for example by keeping a record of students I've called on to answer a question in class and being sure to call on students with consistent frequency. I could then review my records periodically to be sure I'm being consistent. More subjectively, I think I will need to use assessment (both formal and informal) to inform me on whether I am giving struggling students the sort of personalized instruction and motivation that they need to succeed in my class.// __Does equitable class participation mean that all students participate the same amount or the same way? __ //As we saw in the one of the videos in class, there was an unconcious gender bias in the classroom, I believe showing the allotted time for a female student to respond was much lower than a male student. In regards to the above question, we should give the same/equal response time, but what if the student's answer is not sufficient? Of course, the teacher gave equal opportunity, but the answer was not correct. Does this student require more participation so the teacher ensures the student is learning (which might reflect gender bias) or is equal participation sufficient? // __How can I as a teacher ensure that when treating my students equitably, that they still believe that I am being "fair"?__ //Since being equitable can mean giving the students who need more help, more of my time, it is important to show the students that no one group is favored over the other. Perhaps by getting them involved with helping the students that need extra help would be beneficial as it shows them that some struggle with the material more than themselves, as well as freeing up more of my time to help others.//

===Teaching Practices that can be used to make high standards accessible for all students __**: **__===

__Teachers should involve parents in their children's education as much as possible.__ //Studies have shown that when parents are involved in children’s schools and education, children have higher grades and standardized test scores, improved behavior at home and school, and have better social skills.// __Teachers should provide their students with prompt feedback on all student work including homework, quizzes, and tests.__ //Giving students feedback on their performance and progress will help students focus their learning, correct misunderstandings, and help enhance their learning experience.// __Teachers should always work to understand the background and culture of their students.__ //By having a better understanding of the "whole" student, the teacher can help figure out the best way of teaching. Also, what other help students might need to be successful in the classroom.// __Teachers should encourage students to succeed in class and provide extra resources such as extra help outside of the normal school day.__ //It is also necessary to find out about the community that the students live in order to adjust expectations. This is very important as students do not always have the support and resources needed to succeed in school.// __Teachers need to avoid the "squeaky wheel" syndrome where the most vocal students are those that receive the most personalized help. Instead, the teacher needs to use regular assessment (both formal and informal) to determine which students are in need of the most personalized help, and allot their time and resources accordingly.__ //In my experience, the maxim that “the squeaky wheel gets the oil” was very true in my math classrooms growing up. Students who openly expressed their struggles got far more personalized help than introverted students who instead suffered in hurt silence until struggling on an eventual quiz or test. In order to avoid this syndrome, I need to be sure to check in with quiet students just as frequently as more vocal ones to check on their progress. This may mean taking a closer look at their homework or offering them opportunities one-on-one to voice their confusion.// __A teaching practice I can use to allow all of my students to achieve high standards is using pre-assessments prior to the start of the course and using assessments throughout the course to monitor the progress of my students.__ //After assessing my students I will be able to determine who needs extra help to meet the standards. If several of my students perform poorly, I will know that I will have to examine my teaching methods and modify them to reach all students.// __Teachers should get to know their students at a personal level to better understand their students and establish a stronger foundation for the “discipline system” and academic work.__ //If the teacher makes an effort to get to know the student the student will tend to recognize the effort and will be encouraged to interact, thereby increase their opportunities to learn. “Begin listening to the hearts of the students, instead of simply challenging their minds”. Make relationships a priority. Make a connection with your students. The gesture could be something as small as finding out your student’s birthdays and being one of the first to wish them a “happy birthday”, or making a concerted effort to memorize the student’s names earlier than expected. Meeting with students in private conferences talking academics or about extracurricular activities will might show them that you are making an effort to get to know them and they will acknowledge you for that effort.// __Make it relevant. If a student does not see the point of learning the material then they have little chance of meeting high standards. __Based off what I have learned, a student is more willing to try and meet high standards if they can complete the phrase "If I do ___ then I can achieve my goal of__ ___"__

**__Overview__:**
A curriculum is made up of scope, what content is to be taught; sequence, order in which skills and concepts are introduced; and pedagogy, (to some extent) how is content taught and assessed. The purpose of having a curriculum for teaching mathematics has four foundational ideas:
 * 1) The content being taught is organized with tasks that are more worthwhile
 * 2) Creates a dynamic discourse for the students
 * 3) Creates a challenging and supportive learning environment
 * 4) A teacher using a curriculum continually seeks how to improve their teaching.

//Assessment based://

 * 1) Educational goals
 * 2) instructional objectives (unit)
 * 3) Student objectives (lesson)
 * 4) Assessment (Formative-level of development mastery) (Summative-Evaluation)

//Standards based://

 * 1) Standard
 * 2) Benchmark (usually grade level)
 * 3) Learner Outcomes (course objectives)
 * 4) Outcome indicators (Formative/Summative assessment)

As NCTM says:

Psychological and educational research on the learning of complex subjects such as mathematics has solidly established the important role of conceptual understanding in the knowledge and activity of persons who are proficient. Being proficient in a complex domain such as mathematics entails being able to use knowledge flexibly, appropriately applying what is learned in one setting to another. The union of factual knowledge, procedural proficiency, and conceptual understanding enhances all three components, making the resulting learning usable in powerful ways."
 * "A curriculum is more than a collection of activities: It must be coherent, focused on important mathematics, and well articulated across the grades.**
 * Focus and coherence:** Mathematics consists of different topical strands, such as algebra and geometry, but the strands are highly interconnected. A coherent curriculum effectively organizes and integrates important mathematical ideas so that students can see how the ideas build on or connect with other ideas, thus enabling students to learn with understanding, develop skill proficiency, and solve problems.
 * Important mathematics:** A mathematics curriculum should focus on mathematics content and processes that are important and worth the time and attention of students. Mathematics topics may be important for different reasons, such as their utility in developing other mathematical ideas, in linking different areas of mathematics and in preparing students for college, the workforce, and citizenship.
 * Articulation across grades:** Learning mathematics involves accumulating ideas and building successively deeper and more refined understanding. A school mathematics curriculum should provide a road map that helps teachers guide students to increasing levels of sophistication and depths of knowledge. Such guidance requires a well-articulated curriculum so that teachers at each level understand the mathematics that has been studied by students at the previous level and what is to be the focus at successive levels.
 * Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge. Learning mathematics with understanding is essential.**

NCTM Professional Standards for Teaching Mathematics (1991, p. 32) The teacher is responsible for shaping and directing students' activities so that they have opportunities to engage meaningfully in mathematics. Textbooks can be useful resources for teachers, but teachers must also be free to adapt or depart from the texts if students' ideas and conjectures are to help shape teachers' navigation of the content. The tasks in which students engage must encourage them to reasons about mathematical ideas, to make connections, and to formulate, grapple with, and solve problems. Students also need skills. Good tasks nest skills development in the context of problem solving. In practice, students' actual opportunities for learning depend on the kind of discourse that the teacher orchestrates.

[] This link will take you to the National Council of Teachers of Mathematics where you will be able to find more information on curriculum and curriculum ideology. [] This link will take you to a site that has research directly related to curriculum as well as some ways to actually implement that information.
 * __Resources__:**

Questions teachers should ask themselves when making the mathematics curriculum stimulating for their students:
__How will I take the school's curriculum and adapt it to be meaningful and engaging for students?__ The students may not be up to par with the school's curriculum and may need extra instruction to catch up. __When I am hired in a district that requires me to "buy into" a program that I do not believe in, or do not feel that the curriculum is a best fit for my students, how should I (or rather; should I at all) adapt curriculum to fulfill requirements and effectively teach?__ //I am worried about this as I have had the opportunity to observe and teach in multiple districts with varying degrees of confidence in programs. One of the most common problems that I have noticed is that school boards and superintendents have a short sited view of mathematics education. The main goal is to get students caught up to level now; but do little to prepare students for future classes. While creating a strong foundation is important-research points to higher order thinking skills as the key to success in lifelong learning// __How can I, given a curriculum and make it engaging for the student?__ //Some time you are going teach a part of a lesson that you may feel is not that interesting. That makes it even harder to make the lesson engaging for your students.// __How can I construct homework assignments that encourage completion, critical thinking, and can not be easily copied from others?__ //When students complete their homework assignments they are more likely to be successful when it comes to tests as well as they are gaining study skills that will be very beneficial as they continue on their academic career. Motivating students to complete their homework is critical and may require smaller assignments so they don't get overwhelmed or creating fun assignments that interest the students.// __How can I generalize and integrate multiple fields of mathematics while still adhering to the curriculum?__ //Even with strict curriculum guidelines teachers have some opportunity in class to present their students with a bigger picture of how the topic of study applies to other mathematics. Showing students the context of their work on a grander scale may open up doors for multiple representations of a topic (geometric representations of algebra) and a deeper understanding of both subjects.// __How can I teach a curriculum that I do not feel is effective for student learning and how can I adapt it to fit my teaching style?__ //There are many different curriculum systems available and unfortunately, the teacher has little say as to what's used. However, that is not to say that the teacher is powerless to what is being taught. Even if you don't agree with the curriculum assigned, your passion for teaching and subject matter should come out. Also, most mathematics curriculum isn't so narrow minded that you cannot put your own "spin" on it to fit your style.// __Can a teacher integrate other materials (i.e. another math book not currently used) in order to expand on the given curriculum?__ //This could be useful in order to make a curriculum that you do not like more beneficial for both the teacher and the students learning. There are times that 2 different books on the same topic can be very different in approach.// __//As a new teacher should you be more conservative and “go with the flow” when it concerns the mathematics curriculum? //__ //The curriculum has been established by experienced professionals who know more about the content, scope, sequence, and pedagogies necessary for a good curriculum. As a new teacher its actually nice to have an established curriculum as a base/guide and concern oneself more about having an effective and engaging pedagogy and proper classroom management. Most likely as a new teacher there will be other math teachers in your subject areas who you can end up coordinating the curriculum with. That kind of teamwork will permit you to concentrate on lessons and making them fit into an established curriculum. //

Teaching Practices that can be used to make the mathematics curriculum stimulating for their students:
__Teachers should always be reassessing their curriculum to make sure it’s objective is relevant and being accomplished.__ //Teachers who have a set of lesson plans that have always gone smoothly so they reuse it every year without analyzing whether its objective is still relevant or if the students aren’t being stimulated.// __Each teacher should be cooperating with each other in order to ensure that each student is being taught the fundamentals so that they may move onto higher levels of math if desired.__ //Time is a very important factor when working with students who might take a little longer to learn what is being taught. It is ideal that as a teacher you allow time before and after school to help the students who may have questions.// __Teachers should provide students many opportunities to participate and be engaged in the lesson.__ //Students don't learn much by simply listening to teachers lecture and memorizing formulas. Students need opportunities to discuss new material, relate it to previous experiences and knowledge, and apply it in a real world context.// __One area of mathematics education I find particularly exciting is the current innovations and legislative action in STEM education.__ //We are fortunate in Washington State to have an amazing supply of tech companies that are partnering with school district to help create curriculum that is engaging and real world applicable. We should use these opportunities to benefit our students and help them compete in the 21st century job market.// __Teachers should communicate with other teachers in the math department. They may have ideas and lesson that are more exciting and effective for students.__ //Often, other teachers in the math department have had the same students that you currently have. They can tell you what kind of lessons worked for whom. They can help you help your students learn, that is very valuable and a resource you may not otherwise have.// __Teachers need to pull all facets of a given math lesson in and assess what will be effective for a specific class, because it won't always be the same from period to period.__ //This will enable a direct flow from the curriculum to a stimulating environment for each unique class.// __Every book has set scope and sequence, however, most districts also have a scope and sequence which may differ from the book.__ //The key is to address the scope required by the district, analyze your specific classes, and develop a sequence that works best for your current situation. Just because the book has chapters 1-12 doesn't mean you have to go in that sequence, Make it fit your pedagogy.// __Avoid just teaching "plug and chug" methods of solving problems.__ //Show students WHY the formula works when possible and explain why each step is necessary so that they understand what they are doing. Then when they face problems that are formatted slightly differently that can think critically and apply what they know instead of trying to just apply a generic formula.// __Teachers should make themselves very familiar with the curriculum assigned, not just the chapter that the students are working on.__ //It's important to know what areas you need to spend more time on, and you won't know if you're just working page-by-page and having a set schedule. Some topics are more important, and the curriculum direction might not make that completely clear.// __With the Curriculum a teacher needs to be able to adjust the level of rigor__. //Creating a challenging and supportive learning environment with the curriculum is important. If it's too hard students will just give up, and if it's to easy they will not pay __attention.__// __Math teachers should be collaborating with each other as well as their coworkers teaching other math-based subjects such as the applied sciences.__ //This collaboration could open doors for a more hands-on mathematics, giving students opportunities to construct and analyze mathematical models and find the relevance of their studies in the real world.// __Math teachers should be working together to teach the same material.__ //Teachers who teach the same classes need to cover the same material in the same amount of time. Many secondary schools have semester systems where students can have different teachers for each semester. It is important that everything is being taught so that students do not fall behind when they get to the semester break or even to the next class. This ensures that students have the necessary skills required to excel in the next class. That comes from communication within the math department.// __The math department should communicate on a regular basis on the curriculum and how the students are doing in general.__ //The teachers should try to teach the same material in the same amount of time while remembering that some students might need more time than others. Those that need more time should be given the resources and time to complete assignments and be successful in learning the curriculum.// __** Other than coming up with engaging daily lesson plans, a creative and effective pedagogy and a safe and efficient classroom management plan, he best thing a teacher can do to encourage a stimulating mathematics curriculum for their students is to bring some excitement and enthusiasm to the classroom. **__//I’ve read things like having a passion for mathematics and sharing that enthusiasm with the students. As much as I love math it is difficult for me to feel passionate about math. Don’t get me wrong, math has been very good to me. My passion lies in teaching math and the hope of making the difference in as many students lives and hopefully changing their attitude towards mathematics. I believe that some students just need to see they can succeed and their confidence and enthusiasm for math will flourish.// __Teachers should design activities that make clear connections between math concepts, procedures, and problem solving or reasoning.__ Problem solving activities that focus on understanding a concept both procedurally and abstractly always is a good math lesson.

Teaching
On the website “Teaching Math,” Maria Miller states that there are four points of effective math teaching. They are, (1) let the material make sense, (2) have goals for your students, (3) use effective tools, and (4) live and love math. By allowing the material to make sense, we can begin to focus on the “why” something works, not the “how”. Thus, students will gain a better understanding of the math involved. Furthermore, we should have goals set for our students so we aren’t teaching just to get through the book, but rather to create individuals who will be contributing members to our society. Also, using effective tools can improve our teaching. Some of them may include real life story problems, the book, internet lesson ideas, and related math games. Finally, if we are enthusiastic about math the students will be able to feed off of our energy and begin to like math as much as we do. Along with these four points, there are multiple ways to help students succeed in math. For example, making information meaningful can help students recall and retain the material being taught. To further help students recall information learned, anticipatory sets can reiterate what was taught the day before as well as help the teacher become aware of the students prerequisite knowledge before starting the new lesson. In addition, students are given the opportunity to clarify any information that may have been confusing to them. We must also allow students to actively participate throughout the lectures. Not only does this continue to help with retention and recall, but they will grasp a deeper understanding of the subject matter when given the time to openly discuss it. Having these discussions in class takes the emphasis off of merely memorizing formulas and, as writer and award-winning teacher Joe Pagano says, helps “tap into a higher consciousness.”
 * __Overview:__**

**__Resources:__**
[] []

Questions teachers should ask themselves to become a more effective mathematics teacher:
__Can I find a real world application in my lesson?__ //There are times when math class is just that: math. As a teacher, my job is to make math about more than a class or a grade. There should be a purpose behind what I teach and the students need to be aware of this.// __How can I make math more enjoyable for students when society does not instill a love for the subject?__ //To make math enjoyable you have to eliminate the stigma that comes with being in a math class. Many kids will say it's their hardest class and that they will never use it again. It's our goal as the future educators to make sure that they know that persevering in tough times is a good life skill that they will need. Sometimes school isn't about the content, it's about the skills acquired.// __How can I teach to many different levels in the same classroom?__ //It is vital to know what students need more individualized attention and which don't. The teacher can modify assignments if need be to make it more appropriate for the different levels. However, it's important to always keep the other side in mind too; when the students that need extra challenge problems or questions can get them. Whether it's modifying an assignment to make sure the students get the requisite skills down, or modifying the assignment for others to help ensure mastery, differentiated instruction is important.// __How can I build my curriculum to make the math concepts that I'm teaching more relevant to my students?__ This question is important because my students will be far more motivated to solve problems that they're interested in. So, I need to find ways to ask the right sorts of questions and thus engage my "audience. __** How can a teacher demonstrate an enthusiasm for teaching mathematics? **__ I agree that teachers need to show an enthusiasm for teaching mathematics but teachers come in various personalities. For some with extraverted personalities expression comes easy. What can the more reserved teachers do to express that same level of passion they have for math? __Does teaching effectively mean student success in grades or "selling" math to students?__ This is important to ask because student success can be measured in many different ways. Developing an appreciation for math as well as helping the student get good grades in math can be very important. __Are the students able to complete similar problems when presented in different ways?__ Often times students can get by in a math class but just applying formulas in a systematic way. However when students can analyze a problem and apply what they have learn in different ways than it shows a deeper understanding of the material. __Have I assessed my classroom well enough before my lesson that I know what content they will struggle with?__ Teachers should assess their students prior to learning, finding their strengths and weaknesses in the new topic of study. This way teachers can better fit the lesson to the students' zones of proximal development. Note that these zones will vary with each student and each math topic.

Teaching Practices that can be used to become a more effective teacher:
__The teacher should make the classroom fun and exciting as possible.__ This is first done by gaining the respect of the students, but also developing a relationship with them in which the environment is both relaxed, positive, enjoyable, and productive. __Being a teacher is more than just educating students__. You must walk into class with the right mind-set every day. Students can easily detect your mood and will reflect what they see. If you appear unenthusiastic, they will reciprocate. You must maintain a positive, encouraging attitude in the face of all challenges. __Teachers need to be able to effectively engage their students each and every day.__ However, just like eating the same lunch everyday will eventually lower our expectations, a variety of classroom activities and lessons strategies should be used to keep students on their toes and always eager of what may come next. __The most important thing teachers can show their students is belief.__ //You must believe they can accomplish what you set out for them, and they must believe that you actually believe it. If a teacher thinks I can do something and pushes me to do it, then I know I can achieve it. You must also be willing to help them get there. It's one thing to tell a student you believe they can accomplish what you've laid out for them, it's quite different to stay late after school helping them get there.// __Teachers need to constantly check for student understanding.__ Teachers should always be prepared to adapt or change a lesson based on the students' reaction to new material. When the students show a good understanding of the concepts, the teacher can make it more challenging and in-depth. When students are struggling with new concepts, the teacher should slow down, give examples, and be ready to answer questions to make the material clear. __We need to accept that every student will not love math as much as we do.__ But if a teacher is able to create a classroom environment that is safe and accessible for all students, students will learn to appreciate the field and hopefully, math anxiety will be lessened. __When teaching a subject that can be difficult for most students remember that to much, to fast, won't last.__ It takes more time for some students to gain an understanding of the material, so be aware of that when you are reflecting on your teaching style. __//I strongly agree that teachers need to have a passion for the subject they are teaching and that it needs to be expressed through their teaching.// I was personally affected by the ezine article “Effective High School Math Teaching: A Recipe for Success” by__ [|Joe Pagano]. I have a somewhat reserved, low-key personality and it has been a concern of mine that it may be a limiting factor in my future success as a teacher. I have observed a similar type teacher who seems to go into “salesperson” mode (as the article puts it) in front of his students. I totally agree with the article that “if you manifest a genuine concern for the individuals that you teach, mix this with a little humor, present the material with a commanding knowledge, and show enthusiasm for what you are selling, then the victory will be easily at hand.” I've noticed that teachers with the colorful personality has a better chance connecting with their students. Fortunately thats not all that goes into making a good teacher and there are other means to get there. __Math teachers need to teach problem solving and give students the opportunity to practice their skills.__ It is not only important to teach students HOW to solve problems but give them problems to solve. It is important for students to gain confidence in math. Knowing how to solve problems can start the student on the road to success.

**__Overview:__**
Learning is the modification of behavior through practice, training, or experience. In mathematics to understand and learn the concepts students need practice; to gain practice problems and worksheet or homework are ways for this to happen. Learning for understanding in mathematics can only be done with engaging and encouraging students to do math their own way and relating to real world issues. This way a student will retain more information when they can relate to the issue. Along with being able to relate the students need to be shown how to have ownership of their learning as well. When teaching for understanding of learning a student needs to know why they are learning/doing something. Students need to be able to understand the language of mathematics so in explaining the vocabulary and using it as the teacher it will encourage the students to use the vocabulary too. Students also learn in many styles of cognitive thinking (Cognitive/Learning Styles website) · scanning - differences in the extent and intensity of attention resulting in variations in the vividness of experience and the span of awareness · leveling versus sharpening - individual variations in remembering that pertain to the distinctiveness of memories and the tendency to merge similar events · reflection versus impulsivity - individual consistencies in the speed and adequacy with which alternative hypotheses are formed and responses made · conceptual differentiation - differences in the tendency to categorize perceived similarities among stimuli in terms of separate concepts or dimensions <span style="font-family: 'Times New Roman','serif';">[]

<span style="font-family: 'Times New Roman','serif';">When introducing new ideas that students have never seen, teachers can use making sense activities to help them learn. Making sense activities are also known as games. <span style="font-family: 'Times New Roman','serif';">Games Help to promote these ideas:


 * strategize
 * rewards (winning, etc.)
 * engaging with interaction
 * socialize
 * manipulative

**__Resources:__**
<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">This web site talks about learning styles or multiple intelligences in the classroom: [] <span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">This web site is full of many levels and disciplines of fun math games to do in a classroom or extra help activities: []

**Questions teachers should ask themselves when trying to understand how students learn mathematics:**
__How can students learn math when they can't speak English?__ This is a reality. As a teacher, I will have ELL students and will need to adjust my curriculum to include them. Attention to detailed steps and imagery will help these students as well as providing additional, outside resources for their use. __<span style="font-family: Arial,Helvetica,sans-serif;">Do you plan to utilize math games in your classroom? If so, what purpose will they serve? __ //<span style="font-family: Arial,Helvetica,sans-serif;">Math games are great for end of section reviews, or good for weird periods (like HSPE testing days that shorten the period, or late start days). They are a fun way to make sure the students are learning and that they can participate in a change-of-pace activity. // __<span style="font-family: Arial,Helvetica,sans-serif;">How can you make mathematics relevant to your students' lives? __ This is important as students do not see why they have to learn math if it is not relevant to them. __<span style="font-family: Arial,Helvetica,sans-serif;">What teaching practices best engage students in math learning and how can a teacher assess student engagement during a lesson? __ //<span style="font-family: Arial,Helvetica,sans-serif;">This question is important because the average teacher only engages students about 50% of the time, so one of a few pitfalls is occurring. Among those causes, first, a teacher may be using teaching practices that have no potential to truly engage his/her students. For example, handing out a worksheet for students to work on during class (“busy work”) is not going to engage students. It can be tempting to use such teaching practices because they are easier for the teacher, but they rarely have the potential to engage students and thus should be minimized. // //<span style="font-family: Arial,Helvetica,sans-serif;">Second, a teacher may think he/she is engaging students but fail to realize that a given teaching practice is not working. For example, a teacher may have a class where a few outspoken students give the false impression that the entire class is engaged and learning during a lesson, while the truth is that only those few students “get it.” So, informal assessment must be used to check in regularly. For example, a teacher can ask students a few times during class to give a thumbs-up or thumbs-down on whether they understand what is being taught. This can give the teacher a rough idea of who is engaged and learning. // <span style="font-family: Arial,Helvetica,sans-serif;">__Is there a teaching practice that incorporates multiple ways of learning? When you have a classroom with students of multiple learning styles, how do you develop a lesson plan that reaches all students?__ //When you as a teacher are responsible for your students learning, you want to be able to reach them with the most effective approach that encompasses multiple learning styles. Developing a lesson plan catering to a specific learning style and changing which learning style is used, neglects a portion of students. Using this approach it seems that students will learn only 1 out of the 5 lessons, because it was not in their learning style. Thus, which approach is best?// __<span style="font-family: Arial,Helvetica,sans-serif;">What are some creative visual learning strategies that can be used to supplement lessons in a high school mathematics class? __ //<span style="font-family: Arial,Helvetica,sans-serif;">There are three major types of learning styles: kinesthetic/tactile, visual, and auditory. Auditory learning is the most popular in the classroom through discussions and lectures. In mathematics tactile or kinesthetic learning opportunities are very rare. Many students are visual learners and any chance to supplements lessons with visual aids would help make learning more engaging and effective. // __<span style="font-family: Arial,Helvetica,sans-serif;">So, when and how do you correct a student that is learning new concepts? __<span style="font-family: Arial,Helvetica,sans-serif;"> //Sometimes students use intermediate steps that are reasonable when trying to understand a new concept but could lead them astray in the future. As teachers we have a deeper understanding of the subject and often see flaws in arguments presented by the students.// __<span style="font-family: Arial,Helvetica,sans-serif;">What will engage my "21st Century Learner" students so that they want to learn? __ __<span style="font-family: Arial,Helvetica,sans-serif;">What will get students coming into class to forget what happened out in the hall so that they can learn and be on topic? __ //<span style="font-family: Arial,Helvetica,sans-serif;">From observing in different class last year i noticed that this is a big issue. Students are always talk and thinking about what happened last weekend or what they will do the next weekend or after school. But not about whats going on in class. // __<span style="font-family: Arial,Helvetica,sans-serif;">How can I do more than just teach them math in the classroom, we are more than just math teachers? __ //<span style="font-family: Arial,Helvetica,sans-serif;">I believe most people become teachers because they love kids, not that they love math (or whatever their subject matter is). It's important to have a goal to teach them things beyond mathematics. The skills that students can acquire in school last a lifetime, whether it's organizational, social, or the ability to problem solve. I need to make sure I'm teaching them skills as well as mathematics. // __<span style="font-family: Arial,Helvetica,sans-serif;">How will you as a teacher develop students into "patient problem-solvers?" __ //<span style="font-family: Arial,Helvetica,sans-serif;">This is important because students today are simply learning how to plug different numbers into formulas that were handed to them, rather than thinking about why the formula works. Thus, this "plug-and-chug" math is not only boring, but it also does not help the student learn the material. By becoming a "patient problem-solver," the student can develop a deeper understanding of the math than the student who wants to just answer the question and move on. // __How do I teach to accommodate all types of learners?__ All students learn differently. It is important to cater to each and every student's way of learning. Which includes using various types of teaching styles and technology along with other tools to enhance student engagement. __What type of assessment can I use to ensure that my students have found the value of the mathematics taught, as well as assess their vocabulary?__ //While this assessment could be done in a number of ways, it is important to give an open-ended assessment where the students have to explain a real world use for new mathematical topics using the proper vocabulary.//

**Teaching Practices that take into account how students learn mathematics:**
<span style="font-family: Arial,Helvetica,sans-serif;">__Teachers should always take into account previous knowledge of their students and should pre-assess their students before starting a new section or chapter.__ //Since mathematics builds ideas upon previously learned ideas, it is important to have a strong foundation of math skills in order to progress to more difficult and complex ideas.// <span style="font-family: Arial,Helvetica,sans-serif;">__Plan your lessons using your knowledge of the different ways students learn.__ //Use diagrams, numbers, spoken words, and written words throughout your lecture to explain each concept. Also, allow students to practice new concepts individually or as groups.// __Teachers must include additional, outside resources as often as possible__. //Whether it is a simple worksheet, video, power point, or another teacher, we must learn to seek out new resources that benefits students. By diversifying the curriculum, students will have a better understanding of what they are learning and how to apply it in the future.// <span style="font-family: Arial,Helvetica,sans-serif;">__Teachers need to remember that students learn differently than one another and lessons should be created in order to reach each learning style.__ //When using real world examples to make the material relevant to outside the classroom we should incorporate different forms of the same example in order to reach each learning style, and ultimately each student.// <span style="font-family: Arial,Helvetica,sans-serif;">__Because being proficient at math requires a combination of skills ranging from basic arithmetic to complex problem-solving, lessons need to engage both Behaviorist and Constructivist ideals.__ //In order to motivate students to stay sharp on their arithmetic, I would consider having a competition where students complete an assessment of their basic math skills and then take a similar test once in a while and track their improvement. In order to avoid student embarrassment, I would aggregate their scores within groups in the class or across classes, and focus on improvement rather than the actual raw scores.// <span style="font-family: Arial,Helvetica,sans-serif;">__In my experience, seeing improvement in basic skills through this sort of activity breeds confidence that carries over to more complex problem-solving.__ //Additionally, assessments like this would allow me to identify students that are struggling because of deficiencies they brought with them into the classroom, and I could potentially address those cases individually. Also, adding some level of competition (and a requisite reward) can serve to motivate students collectively to perform at a higher level.// <span style="font-family: Arial,Helvetica,sans-serif;">__I saw this video about a teaching practice that I had to get comments on. It has to do with whole brain teaching and the following is the Youtube link: [].__ //The technique involves stimulating both sides of the brain. I would like to get your opinions. This method is not for me but I would like to know if you think it would be effective.// <span style="font-family: Arial,Helvetica,sans-serif;">__Teachers need to remember why they got into this profession. For most teachers they excelled in math, however we must remember that this discipline is filled with fear.__ //Almost all students believe that they will run into a math class that they will not be able to complete, they they will hit the proverbial "math wall". It is our job to show them that there is nothing they cannot do if they have support and are willing to put in the time and effort. We must also make it known to them that we too are willing to put in the time and effort to ensure their success.// <span style="font-family: Arial,Helvetica,sans-serif;">__Teachers should let their students discover mathematical principles, providing interesting problems to deal with as well as guidance when needed.__ When students discover a formula on their own, they understand the concept behind the computations. Also, students may use different strategies for building the formula depending on their learning strengths.

**__Overview:__**
Educational assessment is the process of documenting, usually in measurable terms, knowledge, skills, attitudes and beliefs. Assessment is a very important part of teaching. There are two main types of assessment, namely formative and summative. Formative assessments are less formal and their purpose is to guide instruction as lessons are being taught. Some examples of formative assessments are quizzes or classroom observation. Summative assessments are generally done as a conclusion to a chapter or unit, such as a chapter test or a large project. According to Wilcox & Lanier (1999), the main goal behind assessment is to consider the students’ thought processes, which is why it is necessary to utilize different assessment strategies. All assessments should be clear and aligned to the class lessons to ensure consistency between what is being taught and what is being assessed.

Assessment is important for being an effective math teacher because it is necessary to determine whether or not students are meeting the objectives as well as understanding the content. For assessments to be effective there should be multiple types of assessments being used. For example, teachers should not rely solely on exams but should incorporate projects and other methods of assessment into the course to accommodate diverse learners. According to Romagnano (2001), assessment expectations should be clearly explained to the students before the exam or assignment is given, and assessment methods should be consistent and as objective as possible. All of these components will help ensure student assessment is effective.

**__Resources:__**
<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">Romagnano, L. (2001). The Myth of Objectivity in Mathematics Assessment. //The National Council of Teachers of Mathematics, Inc,// 94(1), 31-37. 25 January 2010.

<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">OSPI: Washington State’s Student Assessment System <span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">[]

<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">Assessment Matters! Toward Authentic Assessment <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: 115%;">[]

Exemplars: K-12: Standards-Based Assessment & Instruction __http://www.exemplars.com/resources/formative/index.html__

**Questions teachers should ask themselves when planning assessment activities:**
__Is the assessment prejudiced against some students, whether it be language or differences in prior education?__ In order to accurately evaluate all students, we must keep in mind the students' differences and be sure to avoid giving any one person any advantage or disadvantage. __How do I use assessments to make changes in the mathematics classroom?__ This is important as students have very different needs in different classes. If the assessment shows that students are not understanding the concepts the way that the concepts are taught need to be changed. __How can I do an effective formative assessment at the start of the school year to better understand the strengths and weaknesses of my students?__ This is an important question because students will need to be motivated to do their best on such an assessment in order for it to be a useful teaching tool. So, it is not only important to create a good assessment and to know how to use the results, but also how to get students to buy in and put their best foot forward. __How can I make sure that the assessment is actually testing what I want to know? And that the students are getting an appropriate grade for their level of understanding?__ //There are times where assessments don't actually give you the information you're looking for, there could be other factors deciding scores on tests or anything else. Do I allow the to retake it to show that they were having a bad day? I think the easiest solution to this question is to only grade them on the standards and allow them to keep retrying until they've hit the standard.// __Should I use a variety of different assessments? Or should I use the same couple types?__ How do you assess problem-solving skills, reasoning, and communication? Would you implement this through math problems, or could this be done explicitly? The mathematical learning standards require that teachers are teaching these skills along with content knowledge. It seems more difficult to assess “thinking” skills over content. __If assessments are used to determine if a lesson should be adapted, how much should be changed?__ If a only a few are missing the idea do you change the whole lesson or just make minor adjustments? This is important because teachers should know what kind of scores should be used to change or adapt their lessons and when the lesson should be left as is for future use. Would it be helpful for the teacher to implement some sort of method to calm students and prepare them for a test or would that mess up whatever procedure they all ready have? __** How would you rank the effectiveness of these non-traditional assessment strategies: Pre-testing; objective assessment; subjective assessment; self-assessment; interactive assessments; practice exams; group projects; students as audience and peer review; participation? **__ __** How can I assess whether or not a student has the ability to move ahead or possibly skip a math level? **__
 * Much of the effectiveness of the assessment is dependent on the design and planning of the assessment regardless of method. The more assessment techniques used the better the results. It should also be taken into account that students express their knowledge in some ways better than others. **
 * With the math class tracking that exists, often times students are placed into a math class (at the middle school level) that is too easy for them. Than every year as they progress they are participating in a math class that is not challenging enough. Teachers need to be able to detect this early on so that students can take advantage of moving ahead in math. **

**Teaching Practices that collect and use assessment information from multiple sources:**
__The teacher should continually make formative assessments in which they will observe student progress.__ It is also important in the subject of math that each each lesson is building on the previous lesson. On summative assessments, make questions that involve allow the students to analyze and recognize things that they had been taught in previous lessons. __Teachers must be able to look at a formal assessment and be able to discern for themselves, the overall troubles that students are having in the class as a whole.__ Then be able to correct any misconceptions and errors that the class may have. __Teachers should be able to tell what kind of grades his/her students should get on a certain assessment, and know how long it will take them to complete it.__ If what actually happens are significantly different then what you expected, then maybe the assessment wasn't as accurate as you had hoped. Consider two students, Student A who bombs the homework, bombs the quiz, but "gets it" between the formative assessments and the test (assessment that "goes in the grade book"). Now consider Student B who "gets it" on the homework, aces the quiz and aces the test. Student A is penalized for not "getting it" early, so his/her grade suffers. Student B's grade looks outstanding because he/she "got it" right away and earned all of the points along the way. If Students A & B both have the same level of understanding, I believe that the grade reported for these two students should be consistent as well. __Teachers should do all that they can through assessment to understand the knowledge and ability of their students before beginning the curriculum at the start of the academic calendar.__ One Calculus teacher I observed had an outgoing class help her develop a "quiz" for students finishing Pre-Calculus and planning to begin Calculus the following year. The quiz aimed to focus students' attention on critical concepts from Pre-Calculus that would be key building blocks for the year ahead. This both helped reinforce those ideas in the outgoing class, and gave the incoming class a chance to brush up and build confidence for the year ahead. __Teachers should create consistent assessments.__ This allows teachers to track students’ progress. To have a consistent assessment allows the teacher to construct lessons appropriate for the content learned. Also this allows the teacher to make decisions based on a quanitative measure of the student's progress rather than a qualitative measure. __Teachers should double check their assessments to make sure that the questions they ask are assessing the material they intend it too.__ As great as coming up with great elaborate story problems or questons with real world applications are, if the student misses the correct answer because he/she doesnt understand the question, its a bad assessment. Make sure it assesses what you want it to. __Teachers should use formative assessments regularly.__ They give a teacher an idea of how the students are doing and where they might need extra help or time to understand the material. In order to be successful teachers we must always be aware of student progress and always check for understanding. <span style="font-family: 'Times New Roman','serif'; font-size: 10pt; line-height: 115%; margin: 0in 0in 0pt;">__Teachers should be fair and honest about their assessments.__ Not only should formal assessments address solely what is taught, but if the majority of students miss problems regarding one particular concept, the teacher must be fair enough to disregard that question when tallying up the final points. In addition, the teacher must also be honest enough to assess their own teaching using the test results. If time permits, then the teacher should reteach that concept before re-assessing.

Technology
Technology can be a useful tool. There are many different programs that can be helpful in the classroom, though, as teachers it is our job to distinguish when it is an appropriate tool to enhance student learning and be able to anticipate problems of student dependency. As author Mellisa Kelly would say, "Online assessment can be a great way to interact with students." And, as NCTM says, "When technology is used strategically, it can provide access to mathematics for all students." This is what we as teachers need to strive for in the classroom.
 * __Overview:__**

**__Resources:__**
http://712educators.about.com/cs/technology/a/integratetech.htm This is a teacher blog on about.com that discuses using technology in the classroom and what types are effective in her classroom http://www.nctm.org/about/content.aspx?id=14233 This is the NCTM site that goes over their ideals on how and when it is appropriate to use technology in the classroom. @http://www.freetech4teachers.com/ This is a great teacher blog with resources for using technology in teaching.

**Questions teachers should ask themselves when about the appropriate use of technology to teach mathematics:**
__How and when should technology be used in a classroom?__ This is important because over use of technology could hinder a students learning and understanding of material. __How will you determine how much technology you will allow on the tests?__ This is an important question because technology is and will be available for students to use. However, technology can sometimes not truly show whether a student understands the concept or is having the technology do if for them. __ ** What are some of the reasons why teachers have not embraced technology more as part of their math curriculum? ** __ ** Of all my mathematics classroom observations there was only one non-remedial class that had a computer lab as part of its curriculum. The graphic calculator was the only use of technology that I had observed. ** __ ** If teachers are using technology in lessons, should it also be integrated into assessments? If so, how can teachers do this effectively, while holding students accountable? ** __ ** The most common use of technology is a computer. The problem with using these on assessments is the availability of cheating. It is important for teachers to determine how much technology and what types can be used on assessments. ** __ ** Except the calculator, what are some math technologies you can use in your class? ** __ There are virtual manipulatives, in the form of applets, as well as other computer programs. You could also use blogs or wikispaces. The point is to keep the students engaged using technology. __How can you develop a lesson that includes technology that does not get bogged down in the details of how to use the technology?__ Often in the classes I’ve taken, I get frustrated with how to use the new technology, and I miss out on the actual content of the lesson. I think this would also be the true for some of my students.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%;">__ How can a teacher show how technology can be used without creating a dependence on it? __
<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt; line-height: 115%;"> There are lots of examples of how technology is used in the "real world" so it is important for a teacher to show that students need to understand the math behind the techonology, they can't just rely on programs and devices to tell them what they need to know. For example if a person doesn't understand basic math skills, they will not be able to work with spread sheets as effectively even though spread sheets can do most of the math for the <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10pt; line-height: 115%;">students.

**Teaching Practices that use technology to increase student learning:**
__Technology can produce simulations, experiments, physical representations that would take way too long to do in a classroom therefore teachers don’t or can’t do it only being given a short period of time each day.__ All around the world people are creating better ways to connect ideas and technology allows the teacher to tap into them filter and produce an exciting involved lesson for their students. __Teachers must make the best of the technology situations they get put in.__ If the school does not have the appropriate equipment then see if it is possible to receive funding outside of the school. I would also suggest using demonstrations on your own personal computer or calculator to show what the student might be able to do at home, or with a partner if at all possible. __Technology, when used appropriately, can be very beneficial to the students in complete understanding of the material being taught.__ We must be aware that not all students will be proficient in the type of technology a teacher uses and be willing to help them reach an understanding of that technology and how it will help them in learning mathematics. __Understanding that technology is beyond calculators and other computer aid software to make math more simple and faster.__ Technology in the classroom is flip charts on the activeboard with saved students work and questions. Its podcasts or better yet, video podcasts. Its online references to practice problems. Its being able to take student polls with cell phones. Its online assessment to track where errors occured for individuals and trends of a class. All of these things will better our classroom and should be integrated into our daily routine. __Teachers should be willing to make sure that students understand how the technology will help in learning math.__ At the same time the teacher should be aware of situations where technology would undermine the learning of the students. __Teachers should use technology in a deliberate way that enhances students' mathematical understanding.__ An example of raising students' "mathematical thinking" from a single representation to multiple representations that we've explored in class is the use of Logger Pro. This program does a nice job of simultaneously showing a physical representation (the pendulum), a graphical representation (sine curves), and numeric representation (spreadsheet). The program even allows students to eventually fit a symbolic representation by choosing an appropriate equation from a menu. __ ** Deliberately integrate a computer lab into his/her curriculum one day a week or once every two weeks and commit to a technology-assisted lesson. ** __ ** There is no doubt that the computer, when used appropriately in a lesson, is one of the more successful ways to engage students. Not only would students benefit from some instruction variety but they would benefit from some worthwhile knowledge as well. ** __Teachers should explain the use of technology in aiding math discovery so that it does not become a crutch.__ Technology, often times, is abused. Students use it for short-cuts rather than deep understanding. It is important for the teacher to explain that technology enhances learning, not shortens it. __Teachers need to know the limits of technology and not just to use them because they're available and they assume everyone will like it.__ If you have an incomplete lesson then no Smartboard or technology can save you. One shouldn't use technology just because it's available, I think it'd be better to rarely use it but make sure it's impactful and needed. This is my problem with Smartboards. I dislike them. I don't want to be constrained to one board, I want to walk around the class and write on many different boards. I want my examples to be around the room not on many different slides that can't be seen. __Teachers should use technology to facilitate learning and understanding rather than teaching students how to use the technology to solve problems.__ Technology can sometimes be used to teach students how to solve a problem the student should know how to solve on their own. Particularly the graphing calculator can become a crutch for students, and when the state exams arrive, students may not remember how to complete the problem by hand. __Use technology that is part of students' lives, to teach and get the students interested in math.__ //Clearly, there is a difference between having students simply pressing button on a calculator or in a math computer program, and actually exploring the concept with technology. The difference is the problem-solving approach--if students are actively problem-solving, they will be learning and enjoying the math.// ====__Students socialize on Facebook, and have a difficult time living without their cell phones. Teachers would be wise to incorporate their interests in the lesson. However, using technology does not mean that students should only learn how to punch buttons on their calculator, or what button to click on the calculator.__==== Technology can be a great way to reach those visual students whose needs are often not met in the classroom. It is also a great way to engage students and show them the importance of math. A profession in any technological field will almost always require sufficient math skills, so teachers can show their students why they really will need math outside of school. ====__Teachers should use technology for efficiency and accurate modeling.__ While students should understand how the processes of mathematics, technology can make math move at a quicker pace and can reinforce the accuracy of mathematical modeling.====

Getting Started

 * Click on the edit button above to put your own content on this page.
 * To invite new members, click on **Manage Wiki** and **Invite People**.
 * To change your wiki's colors or theme, click on **Manage Wiki** and **Look and Feel**.
 * To set who can view and edit your wiki, click on **Manage Wiki** and **Permissions**.

Need Help?

 * Click on the help link above to learn more about how to use your wiki.