Educational Aspirations

Student Groups and Debates

Teachers often have students work in groups to solve problems. Educators may recite that “two heads are better than one” or something of that sort when talking about the power of effective collaboration. I’ve seen firsthand how student grouping can impact decision making and student learning. How a group interacts will often influence outcomes. Positive interactions between group members often spurs a team to meet their goals. I believe most teachers encourage positive talk during group activities and many set up a norm/expectation list for behavior. Learning is often stretched when students are encouraged to explain their answers to others.

What happens when a student explains an answer and the other party isn’t receptive? Or, what happens when students disagree on an answer or how to solve a problem? This is bound to happen from time to time, but I don’t think this is necessarily a negative. Students should be able to stay on topic and analyze their own argument without expressing frustration towards the idea (not people) that they disagree with. Disagreement may conjure anger if not carefully managed. This requires clear expectations and modeling by the teacher. Easier said than done? Yes. Often “I agree” statements can overshadow academic misunderstandings, while students just follow what the leader is saying in the group. I’m aware that some classrooms encourage debate and I think that in some cases that benefits the classroom. I should also note that having a classroom/group debate depends on the problem and is purely situational.

Students, no matter what their age, need to be able to communicate their ideas in order to meet goals. It’s perfectly fine for students to disagree with the group. How that disagreement is communicated and received charts the course for the group. Individual insights hold value and each contribute to the overall goal of the group. Students need to be able to disagree respectfully, but understand that the team is working towards the same goal. Students that have this mindset are able to offer differing opinions, but innovate as a team.

Having a balance is key. Groups should work together but also be open to differing ideas. Disagreement often forces other students to justify their positions. Justifying provides opportunities for students to analyze their own argument, which gives the teacher a better understanding of a student’s understanding of a particular topic/concept.

I think this also plays a role in how adult teams operate as well (see Ringelmann). I’m going to end this post with a quote from James Surowiecki, the author of The Wisdom of Crowds.

“The wisdom of crowds comes not from the consensus decision of the group, but from the aggregation of the ideas/thoughts/decisions of each individual in the group.”

Picture Credit: S. Miles

Math Debates in Elementary Classrooms

Over the past few months I’ve dedicated a good amount of time to to having math conversations. These math conversations occur when the class is unsure of how to solve a problem or when disagreement ensues over what particular strategy should be used to tackle a problem. The math conversations (or debates) allow students the freedom to openly discuss logical reasoning when solving particular problems. These conversations can be sparked by the daily math objective or follow another student’s response to a question. It’s not necessarily planned in my teacher planner as “math conversation” in yellow highlighter, but I do make time for these talks as I feel that they bring value and encourage student ownership. The conversations also give insight to whether students grasp concepts and are able to articulate their responses accordingly. Mathematical misconceptions can also be identified during this time.

During these conversations I have manipulatives, chart paper, whiteboards, iPads and computers nearby to assist in the discovery process. I emphasize that there’s a certain protocol that’s used when we have these discussions. Students are expected to be respectful and listen to the comments of their classmates. To make sure the class is on task I decide to have a specific time limit dedicated to these math conversations. Some days the conversation lasts 5 minutes, other days they may take upwards to 15-20 minutes. When applicable, I might use an anchor chart to display the progress that we’ve made in answering the questions. I should also mention that sometimes we don’t find an answer to the question. Here are a few questions (from students) that have started math conversations this year:

Why is regrouping necessary? (2nd grade)
What can’t we divide by zero? (3rd grade)
Why are parentheses used in math? (3rd grade)
Why do we need a decimal point? (1st grade)
When do we need to round numbers? (2nd grade)
Why is a number to the negative exponent have 1 as the numerator? (5th grade)
Why do you have to balance an equation? (5th grade)
How does the partial products multiplication strategy work? (3rd grade)
Why do you inverse the second fraction when dividing fractions? (5th grade)
Why is area squared and volume cubed? (4th grade)

Above is just a sampling of a few of the math conversations that we’ve had. Afterwards, students write in their journals about their experience finding the solution to the problem.

Of course this takes additional time in class, but I believe it’s time well spent. The Common Core Standards focus on depth of mathematical understanding, rather than breadth. This allows opportunities to have these conversations that I feel are beneficial. They also emphasize the standards of practice below.

CCSS.Math.Practice.MP1 – Making sense of problems and persevere in solving them.
CCSS.Math.Practice.MP3 – Construct viable arguments and critique the reasoning of others

Photo Credit: Basketman

Do you have math conversations in your class?

Standardize This

Education reform continues to make headlines as US student achievement is compared to the achievement of other countries. An overall increasing focus on standardized assessments has been at the forefront of many of these reform discussions. Teachers and school districts often get caught in the middle of these types of discussions From what I’ve observed, what seems to agitate some educators is the notion that one high-stakes standardized assessment can validate/invalidate the success of a school year. Even though educators have been critical of this notion, federal, state, and local school boards continue to look at standardized assessments as the go-to for quality control/accountability purposes. I truly feel as though these boards have good intentions, but I would like to encourage them to look at alternative ways to measure school achievement.

I don’t know a teacher that doesn’t believe in accountability. Teachers inherently feel a sense of accountability for their students. The way that accountability is being measured and the consequences that occur if growth isn’t met is what’s causing concern. Critics emphasis that only focusing on standardized test scores encourage teaching to the test, massive amounts of test prep and unfortunately cheating. I’m not downgrading the value of standardized assessments as I believe a limited amount are beneficial in providing valuable feedback that can inform instructional decisions. Appropriately utilizing student assessment results may prove beneficial for a teacher or school, but using that data outside of its context to manipulate accusations can cause problems.

Proactive Steps …

By now most educators have realized that student achievement data is starting to make up an increasing portion (20% + ) of one’s evaluation. In some cases one VAM assessment could be used to measure student growth and impact employment decisions. Instead of using one standardized assessment to determining teacher effectiveness, administrators should enable teachers to show student learning through a variety of means. This is a difficult task to tackle as administrators are also being assessed on standardized assessment results. While one assessment shows a singular brush stroke of learning, the picture becomes much clearer when multiple data points are used. Even NWEA, the makers of the MAP assessment encourage school leaders to use multiple data points (not just MAP) to measure student growth. Regardless, some districts are already using singular assessments for evaluation/employment purposes. I’m advocating that principal’s take a closer look at multiple student achievement data points instead of relying on one growth indicator.

How …

Formative assessments, student projects, presentations, and pbl activities can show learning at varying levels. This collection of student data can not only help inform instructional decisions, but show evidence of student learning. Digital portfolios are making a splash in education and I’m hoping that more districts start using them in conjunction with standardized assessments to provide evidence of student learning. Showcasing student learning through a variety of formative assessment tools gives more meaning to the learning that’s happening. If communicated appropriately, state and local schoolboards will take notice and become more interested in multiple data points to determine effectivenessss, rather than a singular one.

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Equivalent Fractions Tweak

A few days ago I started gathering resources to supplement a math unit on fractions. The classroom was studying equivalent fractions and I thought there might be a variety of resources available on a few of the blogs that I regularly visit. I generally follow the #mathchat hashtag and find/share ideas that relate to mathematics. While reading a few math blogs on fractions, I came across John Golden’s site that has some amazing ideas that can be used in math classroom. His triangle pattern template sparked my interest.

John provided a template that’s available on his site. I printed out the template and began filling out each triangle with fractions. I ended up with a sheet that looked like this.

So what happened?

First a lot of brainstorming and error checking. Then I decided to have students cut out the triangles and compile equivalent fractions. This is what happened …

Students in fourth grade cut out each triangle and combined them to make equivalent fraction squares. Students worked in collaborative pairs during the project. I observed students using math vocabulary and having constructive conversations with each other to finish the assignment.

Before giving the assignment to a fifth grade class I decided to eliminate two triangles on the sheet above. It was the job of the student to find what triangles were missing and create equivalent fractions to complete the squares. The students were engaged in this activity from start to finish. Some students even wrote the equivalent decimal next to each square.

Overall this project took approximately 45 minutes to complete and it was worth every minute. Students used the terms fraction, improper fraction, mixed number, numerator, denominator, multiplication, division, and pattern throughout the project.

Just as I did, feel free to tweak this project to best meet the needs of your students.

How Do You Measure School Achievement?

Student achievement is often at the forefront of the minds of school leaders. Achievement, in the form of student data, inundates schools, teachers, and school improvement plans. Moreover, school improvement plans seem to hiccup when state assessment results are released. This data is generally looked at in the form of numerical values related to student achievement on state standardized assessments. If utilized correctly, analyzing student data may be beneficial for a school/district. Reviewing strengths and concerns may help allocate resources to areas of need. Regardless of the benefits, some school leaders and teachers often focus on the need to raise test scores in order to receive a positive rating from the state or district. That pressure can lead to anxiety and fear as student data is now being tied to teacher/principal evaluations.

I believe school achievement can be viewed through a variety of lenses. Even though the nation doesn’t necessarily measure the traits below, I feel that they are still valued.

Character

How well does your school emphasize character? Some schools use programs, such as Character Counts, while others have deployed their own character curriculum. Discussing anti-bullying strategies and positive ways to cope with problems can be contained within this category. Emphasizing positive character traits can contribute to becoming more responsible in and outside of the classroom.

Reflective Thinking

Are students allowed time to reflect on their own learning? Students that are given opportunities to review their achievement/behavior/study skills may set appropriate goals to better themselves. This skill will be used throughout their life. Is goal-setting important in your school? Can students analyze their own achievement and create appropriate goals? Setting and achieving goals often brings confidence and focus. Goal-setting can be emphasized at any grade level.

Creativity

Does your school allow time for students to be creative? Does your school offer opportunities for students to be curious? Are student assignments geared to offer one solution or multiple solutions? I’ve found that curiosity often leads to creative thinking. Creative thinking is difficult to measure, but the value is immense.

Requesting Feedback

Allowing feedback gives opportunities for stakeholders to offer input regarding changes that may benefit the organization. How the feedback is utilized may possibly advance a classroom/school/district’s vision. Using the feedback to better an organization shows value. Data in the form of feedback can be just as valuable as standardized assessment data.

Collaboration

Does your school encourage collaboration? Collaboration between staff, or collaboration between students? I would think it might be both. Is the classroom environment conducive to collaboration? Working together also encourages students to practice positive character traits.

I understand that not all of the skills above are measurable Understanding that school achievement goes beyond standardized assessment results is a step in the right direction.

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Teaching Algebra Through a Different Lens

I recently taught a lesson on pan-balance equations. In my curriculum pan-balances are taught as a precursor to more in-depth pre-algebra. My students seemed to understand simple pan-balances and found that the balances (like an equation) needs to be balanced to work. The majority of students had no problem with questions (like these) involving oranges, apples, paperclips, etc.

Day Two

During the next lesson I introduced the idea of variables with equations, like 2x + 4 = 18. Students seemed to understand, but less than the first lesson. I brought everyone up to the classroom whiteboard and practiced many problems with the students. The students who understood always seemed to raise their hands, while students who didn’t completely grasp the concepts tried to blend in with the carpet. Students became less interested in what I was teaching when I started writing equations on the whiteboard. Unfortunately, I felt like I was losing a battle here. The more the students seemed to not understand, the more I felt the need for direct instruction. Near the end of the lesson around half of the students seemed confident to proceed to the next algebra lesson – solving for x on both sides of the equation.

I had to change something.

Day Three

During the next day I decided to change up my instructional approach. I remembered back to when I first learned algebra and the confusion that I used to experience. My school memories of algebra started and ended by watching a chalkboard and overhead projector, as my teacher wrote and erased equations on the board. This was the only way to learn algebra, or so I thought back then. Using my experience I decided to change the instructional medium.

I started my next algebra lesson with a quick review of pan-balances. Students seemed to gain confidence as we had a conversation about the importance of using algebra in careers outside of the classroom. We watched a quick BrainPop video on algebra and it’s uses. Instead of using the whiteboard again, I decided to take out the iPads. I already downloaded an app called Hands-On Equations a few weeks ago. The students were quickly motivated and I modeled how to use the app under the document camera.

The class and I went through a few problems together until I thought they were ready to proceed. I allowed the students 20 minutes to explore the app and lessons. The students were expected to complete at least three lessons and reflect on their experiences in their math journal. What was interesting was that the students immediately took control of their own learning and utilized the app at their own pace.

Payoff

After approximately 20 minutes, I asked the students to write in their journal how they felt about their journey with pre-algebra. The majority of responses were positive …

“I now understand why we use algebra”

“I never thought algebra could be so much fun”

“Having a picture of the balances helps me understand the concepts better.”

The Takeaway …

After hearing their responses and reflecting on the outcomes, I’m becoming more motivated to vary instruction to better meet the needs of my students. Varying the instructional approach can give students multiple opportunities to grasp concepts that can be particularly challenging. Your students may benefit from a bit of instructional change from time to time.

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What do Schools and Libraries Have in Common?

I recently retrieved a reserved book from a nearby public library. I’m amazed at how libraries have changed over the past few years. Some of these changes have been welcome and encouraged by communities. The notorious Dewy Decimal system has been simplified with an online catalogue, silence is now not expected in all parts of a library, librarians have become “media specialists”, online research is becoming the norm, etc. I could go on and on about the changes in the local library, but I’ll stop here.

This particular library that I went to visit has rows of computers, an ebook reading zone, two copy machines, reservation rooms, DVD/CD check out, moveable furniture, and a ton of space for collaboration. The library has recently been renovated to improve access for patrons. There are study corals for people who need to work independently and reservation rooms for group discussions. A technology corner includes headphones and laptops available for checkout. I could tell that the designers intentionally created certain parts of the library for group collaboration. Those zones have electrical outlets, moveable furniture, and wifi access so that work can be optimized. After leaving the library I started pondering to what extent the library design could be replicated in a classroom setting.

Just as the library environment encourages literacy, the classroom environment plays a pivotal role in limiting/enhancing how students are able to collaborate and learn. I started to reflect on the questions below.

How does your classroom environment enable or encourage collaboration?

Is there an area that is designed for independent work?

Is technology easily accessible for student use?

Is there an area that is designed for research?

I believe that it’s important to ask the questions above and seek answers. The classroom environment should promote collaboration and give students opportunities to learn in different ways. Moving beyond the single row classroom setup encourages students to take ownership of their learning. Giving students opportunities to learn without desks or in a student centered environment may prove beneficial, as some studies are beginning to indicate. Allowing a bit of flexibility in a classroom setup can reap dividends later on in the school year. If the library can reinvent itself, I believe a school/classroom can do the same.

photo credit: j l t cc
Liz508 cc

Dice and Math Computation

Since the beginning of the school year I’ve been searching for different ways to incorporate guided math in my classroom. Guided math has many benefits although organizing the groupings can bring a few challenges. Guided math looks different depending on how the teacher implements the structure. For example, one math group might be working with the teacher while two other groups are using math games or participating in problem based learning activities. The groups will rotate according to a specific time schedule. I’m finding that groups that are not with the teacher need specific instructions and expectations.

For the past few months I’ve been using dice games to emphasize number sense skills. These dice games have peaked student interest and work well in increasing computation fluency. I decided to collect multiple formative data pieces to validate whether the dice games were contributing to student success. By analyzing student data and observing over a period of time, I found that students were becoming more fluent in adding, subtracting, and multiplying small/large numbers.

The games have worked for me, so I’m passing it along to others that might find it useful. Needed materials and pdf files are below.

Materials

A variety of dice (6, 10, 20, 30, etc. dice) Here are some examples:

Templates (in pdf form)

Roll to 150 (multiplication)

Roll to 125 (addition)

Roll to 100 (addition)

Roll to 45 (addition)

Roll from 50 (subtraction)

Roll from 95 (subtraction)

Roll from 35 (subtraction)

Student Data and Balance

Teachers in K-12 often use student data on a regular basis. Student achievement data can be used to qualify students for reading, gifted, remedial, enrichment, acceleration, differentiation, and a variety of other services. Recently, standardized testing data has been the forefront of educational trends and in the news. Implementing a balanced approach when looking at student data can keep stakeholders (educators and administrators) grounded in an understanding that the numbers behind the tests may give light to areas of strengths/needs.

Data isn’t evil

Assessing a student’s understanding of a specific concept isn’t necessarily a bad thing. In fact, over the past few years I’ve grown to appreciate and utilize student achievement data more and more. Whether the data is from a standardized test or not, the data can be helpful if used correctly. Moving data beyond just a number can benefit teachers and students. Data can help teachers ask better questions and provide opportunities to reflect on how students learn best. Involving students in analyzing their own data can encourage student goal setting and ownership.

Having conversations with students about their data is powerful.

Have the conversation

I’m definitely not an advocate for having additional standardized tests, although some seem more useful than others. I find that assessments that give detailed feedback (e.g. areas that need strengthening, %ile compared to the norm, strength areas, next instructional steps, etc.) are more frequently used by teachers, compared to assessments the give little feedback. Obviously, there isn’t a perfect test available for school purchase. The assessments that a school uses should give detailed feedback that can be immediately used.

Do you hear a lot of negative talk in regard to standardized assessments? Having a conversation about an assessment’s effectiveness in informing instruction may be needed. Instead of trash talking the assessments in general, educators and administrators should find assessments that work for them. PLC teams should emphasize the importance of using formative assessments regularly. I’ve found that teacher created formative assessments are some of the best ways to find areas that need strengthening and to identify differentiation opportunities. The purpose of giving the assessments should be communicated to all stakeholders. When teachers understand why the tests are given, (not just for VAM reasons), they may start to value the benefits of assessing students using a variety of tools (such as Common Core performance assessments).

Balance is needed

With teaching and in life, balance is needed. Teaching is a profession that can be stressfull. It has many teachers thinking right now, how many days till Spring Break?? Balancing assessments with instruction takes skill and patience. Standardized tests are often at the forefront of school administrator’s minds. One test shouldn’t be used to determine if success, or enough growth has been made to call that school year/class/school successful. Take a breath and look at assessments from a macro lens. A combination of formative, informal, formal, review checkpoints, activators, performance (insert your assessment here), and even standardized assessments have their place in a school and can be beneficial to a certain extent. The value of the data often depends on how it’s utilized.

Picture Credit: DigitalArt

“Your assumptions are your windows on the world. Scrub them off every once in a while, or the light won’t come in.” Isaac Asimov

Web-Based Formative Math Assessments

I’ll admit it, I’m becoming more of a formative assessment advocate this year. I believe that formative assessments have a place in the elementary math classroom. As a technology enthusiast, I’m always searching for ways to improve my instruction through the use of technology. For the past year I’ve had the opportunity to use Socrative and Scootpad apps (both free) with my math class. Both of these apps are web-based and offer the ability to provide immediate feedback to the student. I’ve added a few snippets of information about these apps below.

Socrative is a web-based program that is similar to a wireless clicker system, but with a keyboard. Teachers can create multiple choice, true/false, and short answer quizes with this app. The quizes are quick and easy to create – I actually created a 10 multiple choice question quiz on an iPad. Teacher have the option for students to complete the quizes at their own pace or at an assigned pace as a class. Similar to Google Docs, student information is updated and you can actually show the data on an LCD screen live. Once short answers are submitted students also have the option to vote for the answer they feel is best. This option definitely promotes student engagement. Reports on student progress can be sent to you via email and they are in Excel format for easy sorting.

Scootpad offers teachers a way to assess students on Common Core Math Standards (Grades 1-5). Teachers are able to individualize assessments based on the needs of their students. Mastery (as a %) can be determined by the teacher and students have opportunities to earn badges and other awards. The interface takes a while to get used to, but overall this app allows teachers a quick opportunity to assess students’ understanding. Student data is aggregated and can be sorted easily. Scootpad will be expanding to middle school math Common Core Standards in the near future.

What formative assessments do you use?