Model and Solve Equations Using Manipulatives:

Using manipulatives is the only way some students can truly understand mathematical concepts. Unfortunately, a prevailing myth in secondary education, especially high schools, is that manipulatives are for elementary children only. I am a strong advocate for using manipulatives and concrete representation for teaching math to students at the secondary level, including high school students.

I was always saddened when a new math teacher attempted to use algebra tiles to teach algebraic concepts and then gave up the effort because of negative feedback from either the students or math colleagues. New research in the field promoting a three-pronged approach to teaching math at the secondary level is encouraging. The Concrete Representation Abstract (CRA) approach is a breakthrough in secondary thinking.

Research Background

Using manipulatives provides students a meaningful context for mathematical knowledge and helps them understand fundamental relationships associated with the. Multiple embodiments – the use of many different models – allow students to focus on common characteristics and generalize to the abstract.

In regards to secondary math, such as Algebra, Henri Picciotto, a mathematics specialist and former teacher writes:
“Even though they cannot make algebra easy, manipulatives can play an important role in the transition to a new algebra course:
• They provide access to symbol manipulation for students who had previously been frozen out of the course because of their weak number sense.
• They provide a geometric interpretation of symbol manipulation, thereby enriching all students’ understanding, and making a powerful connection to another part of mathematics.
• They support cooperative learning, and help improve discourse in the algebra class by giving students objects to think with and talk about. It is in the context of such reflection and conversation that learning happens.”

There are four main commercial versions of algebra manipulatives. In order of their appearance on the market, they are Algebra Tiles (Cuisenaire), the Lab Gear (Creative Publications), Algeblocks (Southwestern Publishing), and Algebra Models (Classroom Products). All four provide a worthwhile model of the distributive law. However, note that only the Lab Gear and Algeblocks allow work in three dimensions.

Bradley Witzel, co-author of the book Solving Equations – An Algebra Intervention, is also an advocate of using manipulatives to teach math through algebra. He describes the Concrete Representation Abstract method. The CRA sequence of instruction consists of teaching students to solve mathematics problems through three levels of instruction, from the manipulation of concrete objects to learning through pictorial representations to finally solving equations through abstract notation.

According to Witzel, the CRA approach to teaching mathematics has proven to be beneficial to secondary students with math difficulties, from small group settings to whole-class instruction. In fact, after receiving CRA instruction, students with learning disabilities had a success rate two to three times higher than their traditionally taught peers.

According to Witzel, CRA benefits students with math difficulties because it presents information in a multisensory way: visually, auditorily, tactilely, and kinesthetically. This multisensory approach causes the brain to process the information several times in various formats, making it easier for students to memorize, encode, and retrieve the information later.

In addition, CRA helps students solve abstract problems without thinking fluently at the abstract level by giving them other levels of learning, whether pictorial or concrete, to turn to solve the problem.

Try using the CRA approach in your classroom and watch your students, even those who struggle most with mathematics, begin to succeed. For more information on this subject and for practical ways to implement this strategy in your classroom, read my newest book, RTI for Secondary Teachers.

For example, when we were studying the solar system, I was able to obtain picture books on that topic, as well as books at the college level. I’d make these books available to students in the classroom for their use as supplemental reading, or background material for projects. I was always amazed at who picked up which books. At times, honors students chose a picture book to read and students with special needs chose a college level text. The students seemed to use a variety of the books regardless of their reading level.

Consequently, I’ve always been a believer in providing books at all reading levels, including picture books as supplemental reading for secondary students. When considering both novel and staid examples for lesson plans in this book, the picture book seemed to be an obvious choice. While at first thought picture books may seem too elementary for high school students, it’s amazing how many students – after they make their wisecracks – actually dig in and enjoy working with the picture books; they are an “easy” focus. Furthermore, starting with simple and easy books facilitates the acquisition of the standards based skills.  The research I’ve cited below backs up the benefits of using picture books at the high school level as well as with English Language Learners in the secondary classroom.

Research Background

Picture books used in the upper grade levels may improve student comprehension. In a study by Bridget Robinson at the University of North Carolina, High School students who studied literary terms with picture books were 72% more successful on a test of literary terms than those who studied with traditional means. Students found it easier to understand and recall literary terms when picture books were used as a teaching tool.

Picture books help students create mental models and help readers build schema. They are written at a reading level accessible to most readers with content varying to meet individual needs. They help students understand complex ideas and vocabulary. Consider that although picture books typically are written for pre-K children, they are meant to be read aloud and therefore utilize high-level vocabulary. We are also finding more and more picture books with adult themes.

The shorter length of picture books permits students to practice their reading strategies and enhance their understanding of difficult content. Picture books allow teachers to “bring up issues, problems, and concerns without deluging students with facts and information”. Another advantage to the shorter length of picture books is that student reading and response is plausible within the brief class periods educators face in secondary schools.

For the English language learner, picture books provide a non-threatening tool with visual cues to support English language acquisition. In a research study focused on using picture books and literature based instruction with high school ESL students, Nancy L. Hadaway and JaNae Mundy found that using picture books engaged students in the language learning process. Vocabulary increased, and reading comprehension skills were evident through class discussion, and writing about their reading experience through journaling, poetry, and research presentations.

The Fitzell Acceleration Center™ is not a product that can be purchased; rather it is a concept and process for center teaching at the secondary level.

The benefit of the acceleration center model is that it’s a powerful strategy for:
• Differentiation
• Response to student learning (or non-learning)
• The inclusive classroom
• Addressing multiple ability levels in the classroom
• Correcting student apathy and lack of motivation

Acceleration center teaching is an advanced teaching structure. It can be combined with any other pedagogy or teaching method.
In broad terms, acceleration centers incorporate a ‘station’ in a corner of the room, a piece of the room, a setup in the room, or a crate in the room. Its purpose is to have students focus on one topic.

That topic might address the standards in which your students did not do well on the state test last year. On the other hand, it might be an aspect of your curriculum that you need to repeat for some students who struggle, while at the same time provide advanced material for those students that need to be challenged beyond the standard curriculum.

The goal is to give all your students an opportunity to succeed. It’s also a perfect structure for delivering Tier Two interventions as well as supporting Tier One (the general classroom) and, where possible, Tier Three interventions.

I remember spending an entire week creating a learning center on plants many years ago. Every single night I was cutting out leaves, flowers, and petals. I made hands-on activities using clothespins and construction paper. I spent hours upon hours on this learning center. Friday was the big day. I introduced the center to the students, and explained the instructions. I had that sense of excitement and fulfillment a teacher experiences after having worked so hard to create what they feel should be an award-winning lesson plan. My students finished the center in 45 minutes. All those hours of prep work and it was over in 45 minutes!

Who has the time for that today? My observation is that, generally speaking, teachers do not center-teach anymore. We just don’t have the time to prepare learning centers using this paradigm. Additionally, secondary teachers rarely use centers. They may be used in some science classes, but for the most part, it’s a foreign concept and considered ‘elementary’.

An acceleration center is not the same as an elementary learning center, however. It’s a combination of several “station” methods. If you have ever created a learning center, implemented an Accelerated Reader™ program, or if you are familiar with SRA (Science Research Associates) Individualized Direct Instruction for Reading Mastery, you have worked with different types of centers. The acceleration center takes the best of each of these models and combines them into a model that is appropriate at the secondary level.

Acceleration centers are similar to the concept of a learning center except that teachers do not create a separate center for every unit they are teaching. They don’t have a holiday center, for example, or a math center created for a specific lesson plan. Instead, teachers focus on a curriculum strand taken from their state standard that aligns with the Tier Two intervention needs of students in the general classroom.

The activities in the center range from the very basic skills in the strand (Tier Two interventions for struggling students) to the highest level skills (Tier One activities and strategies for all students) and, possibly even college-level material (enrichment curricula) if students can reach that far.

For students who need Tier Two interventions, the focus of the center could be based on one of their areas of difficulty: finding similarities and differences, understanding cause and effect, foundational concepts required for algebra, etc. The focus of the center is based on assessment data collected by the general education teacher and the RTI team where applicable.

Once developed, each center remains for the entire year, and beyond.
• You make it once and use it all year.
• You prep it once and possibly add to it during the year.
• The only maintenance required is student assessment and re-assignment.

To learn more about implementing Fitzell Acceleration Centers in your classroom read my book, RTI Strategies for Secondary Teachers.

Rubrics :

Rubrics are performance-based assessment tools used to evaluate student performance on a task, a set of tasks, or a learning outcome. Rubrics use specific criteria, in the form of narrative descriptions, as a basis for evaluating student performance. Most rubrics use a tabular format that identifies the level of student achievement, from low-to-high or high-to-low, based upon the proficiency that the student is able to achieve. Rating scales may be numerical, or qualitative, or both.

Exit cards:
Exit cards are a simple assessment tool. Each card will have a set of just two or three questions for students to answer after you teach a lesson. Students answer the questions before the bell rings. It is the last thing they do in class. They must hand the card to the teacher before they walk out the door, hence the name “exit cards.” It’s ongoing, immediate assessment in action. If you have two teachers, you have two people who can assess and group the exit cards. Exit cards (a.k.a. “tickets to leave”) are used to gather information on student readiness levels, understanding of concepts just taught, interests, and/or learning profiles.

Exit cards can also be used to form intervention pairs, triads, and groups. After a lesson, use exit cards to assess student understanding or interest. Keep the items on the cards short and to the point. Keep it simple!

When reviewing the cards that are implemented as an assessment tool, score them with a #1 if the student does not understand the concept, got the answer wrong, or needs re-teaching. Score it with a 2 if the student understands but needs more practice. Score it with a 3 if the student understands the concept and is ready to move on. Then use the cards to group students.

You might put all the #1s together and re-teach that group (or small groups). Put the #2’s together (or in small groups) and give them a practice activity. Put all the three’s together and assign them an enrichment activity or an investigation. Alternatively, you might put a 1, a 2, and a 3 together in a triad to work practice the skill.

Another option for using data from the exit cards is to determine student interest or strengths in a topic or group by learning style.

Additional quick assessments might include:
• High Fluency Phrases located in The Fluent Reader by Timothy V. Rasinski (Raskinski, 2003). Do an Internet search on a paper titled, “Phrases and Short Sentences for Repeated Reading Practice.”
• Everyday Edits are also effective as both an assessment and an intervention. Search for Every-Day Edits at http://www.educationalworld.com

Still, as I travel the country and speak to audiences of teachers at the secondary level, one area in math that is universally challenging is the skill of graphing.

Teachers tell me repeatedly that they have to stop their lessons and re-teach how to read graphs every time they encounter one in a lesson, text, handout, or news article. When I ask how long it takes to re-teach the process, responses vary from 15 minutes to a full class period.

To help students understand graphing and retain the information they learn about graphs, I would advocate that graphing be a skill that is practiced frequently even as a five minute mini-lesson or part of an acceleration center activity.

Research Background: Students will analyze sets of data.  Data can relate to social studies, science.

Graphing predictions prior to collecting data helps students evaluate relationships and establishes a foundation for the application of scientific and mathematical principles, at their level of achievement.  After data collection, students are able to see how their predictions correspond to actual data.

This process leads students to a better understanding of the nature of both science and mathematics. Science and math play a central role in modern culture. This process allows students to see how the two are linked and how these subjects are linked to actual data in every day applications.

By teaching, and then practicing, how to read and create graphs, students will not only be more successful in mathematics, but will also gain a valuable skill that can help them across the curriculum.

For ideas on how to help your students understand graphs, read my newest book, RTI for Secondary Teachers.

I stressed how important it was to write a professional looking letter. I explained that their letter not only represented them, but it also represented the school. We did not want to embarrass the school. The students took the issue and the project very seriously. One of my students decided to go straight to the top. He wrote to then First Lady Barbara Bush. This student, Charles, also happened to be my most challenging student in the class, behaviorally and academically.

On the last day of school I was packing up my classroom. Suddenly, I heard my name shouted in the hallway in an excited chant. “Mrs. Fitzell! Mrs. Fitzell! Mrs. Fitzell.”  I recognized the voice. It was Charles. Eventually, he burst into the room waving a piece of paper. “Mrs. Fitzell, look, look! I got an answer from Mrs. Bush!” He shoved the paper in front of my face excitedly. “Look!” I oooed and ahhhed and praised him for doing such a great job on that letter. We talked about the assignment and how hard he worked on it. I’ll never forget the pride on his face. That is what a teacher lives for.

Research Background

The creation of a mind map in a small group is an active and collaborative learning exercise. Because a mind map captures a specific topic in a nonlinear fashion and incorporates graphics and colors, this exercise also can connect with learners whose style is not well served by traditional linear, text-based materials.

Fairy tales, cultural fables, and legends are good choices to practice this technique because of their familiarity to students. Most youth can relate to these stories and connect the issues, conflicts, and personalities presented in them to their personal lives. These connections support reading comprehension and enhance long-term recall.

In addition, the brevity of this genre allows for repetition and frequency, which is ideal for initial instruction as well as an intervention. Time constraints pose significant challenges to the secondary teacher, especially at the high school level. Utilizing short texts supports the learning objective without requiring significant class time.

Research Background

Reading comprehension is a complex intellectual process that involves two main abilities: understanding word meanings and reasoning with verbal concepts.  Without these abilities, students cannot comprehend and in turn do not internalize what they read. Comprehension happens within a reader’s mind, which means it cannot be measured in a tangible fashion. Comprehension can only be measured by inferring from a reader’s overt behavior. The use of strategies helps teachers infer a student’s comprehension of material read. Summarizing, recall, and inferential reading are all methods of instruction to determine a student’s comprehension of text.

When assessing comprehension within expository text, students must discern main ideas and concepts from text, contrary to literature where students must identify key elements to summarize a story completely. Summarizing materials allows students to connect to material in a personal manner as they have the opportunity to make a connection from personal experiences to the text. A summary allows students to synthesize, analyze, and evaluate material, which develops higher order thinking skills. However, most students lack the ability to write a summary in a concise manner. When summarizing, students need to identify what happened in the story and state the main details of the expository of literary text. Most students tend to rewrite the story and tell their teacher everything that happened to demonstrate that they read the passage. This is not a summary. A summary will summarize what the story was about in a concise and explicit manner. Using graphic organizers and the Somebody-Wanted-But-So (SWBS) strategy encourages students to develop concise summaries of literature.

As a model for successful teaching of math vocabulary, I’d like to share an example I saw effectively demonstrated at Permian High School in Odessa, Texas.

Ms. McAnnelly and Ms. Pettus, co-teachers at the tenth grade level, made vocabulary review part of their teaching practice. One of the techniques they used was to choose a state test question and spend five minutes a class period, usually at the end of class, reviewing the vocabulary in that item. They did not spend any time working out the math. They only focused on the vocabulary.

They presented a test question via laptop and projector on a screen. Each student had an index card. They were to review the question presented and list any words that they did not understand on that index card. Given that the class had a large bi-lingual population, this was an especially important exercise. The teachers collected the index cards as the students finished. They then used the information on the cards to drive vocabulary instruction throughout the semester.

In addition to having students list the words on an index card, they discussed the question with the class as a whole. I was fascinated to learn that the students found the question phrase, “Which of the following best represents…” the most challenging part of the test item.

We tend to focus on math skill in preparation for state tests in math, however, we also need to concentrate on vocabulary. Many students may do poorly on the test because of a lack of vocabulary understanding rather than a lack of math skill.
It’s important to teach math vocabulary to all students. It’s critical for non-responders.

Also, teach students to look for clue words in math word problems.
• Clue Words for Addition: Sum, total, in all, perimeter
• Clue Words for Subtraction: Difference, how much more, exceed
• Clue Words for Multiplication: Product, total, area, times
• Clue Words for Division: Share, distribute, quotient, average

I remember thinking how much easier it was to use than index cards with a text script written on them. It was also much less restricting. I did not feel tied to reading the cards. Rather, I looked at the picture and went from memory. It saved me from the plight of many presenters: that of being tied to a script.

The technique worked so well for me that I started expanding the idea into my teaching efforts. As I read selections from English texts to my students, I drew the events out on paper in map and graphic format. I would often interject silly ditties and exclamations of passion into the effort to make what I was reading to them stick out in their memory.

Given that my students were at the ‘cool’ age of ‘teen’ they would often look at me and say, “You are crazy!” My pat answer was always, “Yes, I am, but you’ll remember this because of it.” Moreover, they did.

Students learn and remember graphic organizers better if they create them out of their own mental images and patterns. As a parent who has spent my children’s lifetimes trying to teach them how to learn, I was very excited when I walked into my daughter Shivahn’s college apartment and found mind-maps, mnemonics, color, etc. all over one of her walls.

Now, it’s not unusual to find mandalas on her door or on her walls, or flash cards scattered about, but this was a huge mind-map made from recycled 8.5 X 11 inch pieces of paper. I had no idea what it all meant, but I do know it helped her to get an A in the course. My daughter co-wrote a book with me, Umm…Studying? What’s That?, so it’s reassuring to know that she didn’t ‘just’ write the book but also used the strategies and shared them with her peers.

Research Background for Mind Maps, Etc.

Many experts agree that developing cognitive maps and using advance organizers increases critical thinking skills.

Long-term memory files information in the brain through patterns, procedures, categories, pairs, and rules. A mind map uses at least three of these five ways to store information, therefore helping the brain remember information better.

A classic mind map begins with a word, phrase, or idea typically placed in the center of a piece of paper. As the author of the mind map expands upon the word or phrase in the middle, the mind map expands to include various ideas that come to mind when considering that center prompt.

Graphic organizers enable the brain to categorize information. A mind map is a non-linguistic representation method of organizing information that enables students to file information away in long-term memory in multiple modes or memory packets.