2013 NCTM Conference: "Technology: A Portal to Exploration and Discovery"

2013 NCTM Conference (4/19/13 - 8:00 am - Kenn Pendleton):

A big theme through the NCTM conference was using questions to prompt students to explore mathematics. I am a big proponent of inquiry-based learning, so I was excited to find other teachers with similar passions. I was able to collect a lot of fresh ideas.

Kenn suggested that we can use technology as a tool to enable students to explore mathematics. He suggested that we as teachers should ask questions which require students to use technological tools (such as graphing calculators) to explore and answer.

Kenn strongly encouraged the use of technology to allow students to explore the connection between graphs and their symbolic representations. This connection is very important for helping students to visualize math and to build a deeper understanding of what mathematical symbols actually represent.

Kenn encouraged teachers to challenge students to create conjectures based on their observations using technology. They can then test these conjectures by contrasting different scenarios and evaluating whether their conjectures are true or not. This can given them a fuller understanding of mathematical principles than the teacher given a list of facts that go in one ear and out the other.

Technology can be a powerful tool for comparing the graphs of different functions. This can help students contrast functions with their inverse, tranlations, and other properties of functions.

2013 NCTM Conference: "Discover New Ways to Make High School Math Meaningful"

2013 NCTM Conference (4/18/13 - 2:30 pm - Tim Pope):

Tim Pope is the curriculum manager at Kendall Hunt Publishing Company. While this made his session a little more of an advertising campaign, he had some good ideas to share as well.

One statement Tim made really stuck with me, perhaps more than any other at the conference. He said, "If we can remind students of a story that kids can remember, we'll always have something to build on." He elaborated on this idea further, suggesting that if we make math real to students and get them involved in the classroom, they'll remember it much better than they would if they memorized a bunch of formulas. They'll remember even better if a teacher makes math fun as well, or allows students to get up and move around, physically participating as well as mentally participating.

Tim got the audience to participate in an activity identical to one he would use with students. We calculated the amount of time it takes for a specified number of people to complete the "wave", and plotted this data in a scatterplot. We then used this data to create a line of best fit and discussed the properties of this line.

Tim advocated the Van Heile structure for building an understanding of mathematical concepts, and gave an example of each.
Level 0: Visual - students are able to visually identify shapes, patterns, etc., but can't explain why. "It is a rectangle because it looks like a door."
Level 1: Descriptive - students are able to describe the properties of shapes and patterns, but are unfamiliar with the ways different properties interact with each other. "It is a rectangle because it is a quadrilateral with four right angles."
Level 2: Abstract - students can more fully describe the ways the properties of shapes and patterns interact with each other and understand what makes each individual group unique. "I know it’s a rectangle if it’s a parallelogram with four right angles."
Level 3: Proof - students can prove why the properties of shapes and patterns are true. "I can prove it is a rectangle if it’s a parallelogram with one right angle."
Level 4: Rigor - students understand a concept so fully they don't need to consider the basic properties of the concept. "I know that is a rectangle - what would happen if I tried to put it on a sphere?"
Tim suggested that we should challenge students to wrestle with higher-level mathematical questions, that they may gain a fuller understanding of the mathematical concepts involved.

2013 NCTM Conference: "Using Teacher- and Student-Made Videos in the Mathematics Classroom"

2013 NCTM Conference (4/18/13 - 3:30 pm - Janet Andreasen, Deborah McGinley, and Zyad Bawatneh):

Janet, Deborah, and Zyad gave a discussion advocating the use of videos in the classroom - created both by teachers and students - for instruction and assessment. This is a concept I had not personally explored, so I was intrigued by the ideas they shared.

Janet suggested that student-created videos can give teachers the opportunity to view students' problem-solving process instead of just analyzing the end result. The common core mathematical practices require students to draw geometric shapes with given conditions, find solutions using technology, and many other "practices" that are as focused on the journey as much as the destination. Student-made videos allow teachers to assess the practices more fully than they may be able to do through informal observations.

There are many different types of software that can be used for screen capturing and voice recording. Jing and Educreations are free software packages that students can use to create videos. Camtasia and Explain Everything offer more features but come with a price tag. The TI-nsspire also comes with video recording features.

Janet and Deborah suggested many uses for videos in the classroom: First, as mentioned above, student-created videos can be used for authentic assessments to help teachers evaluate the process as well as the final product. These videos, if presented appropriately, can provide insight into students' thinking in a way that a test cannot.
Second, teachers can use videos in a "flipped" classroom to provide instruction for students to view at home or to provide extra support. Teachers can use videos to not only speak to students, but they can also use software such as GeoGebra or visual manipulatives to give students visul representations of mathematical concepts.
Third, videos can be used for student-to-student support. Students can share their ideas with other students through videos. Deborah explained that their district uses Edmodo to encourage students to collaborate together and share student-made videos with each other.

2013 NCTM Conference: "Keeping It Real: Teaching Math through Real-World Topics"

2013 NCTM Conference (4/18/13 - 12:30 pm - Karim Kai Ani):
Karim from Mathalicious.com gave an awesome presentation on the value of "real-world" math to engage students in the mathematics classroom. Karim suggested that math has become very distasteful in our society, "a recipe to something no one wants to eat". He suggested that we need to make math "delicious" again by engaging students in math through topics that they find interesting.

Karim suggested that students today look at math as a bunch of arbitrary rules that somebody made up to torture people for thousands of years. On the contrary, students who explore math based in real life will be more interested in how math works and will be more likely to participate and ask questions.

Karim showed attendees some example lessons from Mathalicious.com that teachers can use in their classroom. The first example he demonstrated was using the Domino's web site to graph the cost of each topping on a medium Domino's pizza. The site doesn't describe the cost of a single topping, but through mathematical evaluation, students can find the cost of each topping and predict the cost of a five topping pizza. The teacher can build on these findings to develop the concepts of slope and the slope-intercept equation of a line (where the y-intercept is the price of a cheese pizza, and the slope is the cost per topping). Students can also compare the graphs of medium, large, and small pizzas.

Karim suggested that teachers can use videos, movies, internet sites, and hands-on applications of science, as well as many other applications of math, to make mathematics more real to students. The world is an interesting place, and there are a plethora of interesting questions to ask. Math is the tool we use to explore these questions.

2013 NCTM Conference: "Using Problem-Based Learning Tasks to Foster Reasoning and Proof"

2013 NCTM Conference (4/18/13 - 11:00 am - Enrique Galindo, Julie Evans, and Jason Walton):
In this session several teachers from Indiana shared their journey of using problem-based learning in their classroom. These teachers worked together to create units based on problems that prompted students to respond to a question in such a way that they are actively engaged in the learning process and driven to learn appropriate skills and concepts.

The first example that one of the teachers gave was a PrBL which challenged students to consider how long they would have to grow their hair in order to donate it to Locks of Love. The students were required to research the necessary data, plot a graph of hair growth, and answer the question using mathematics to defend their opinions.
The students had to research the following questions:
What is the "starting point" of their hair now? Where would this point be on a graph?
What is the rate of growth for their hair? How does this affect the graph plotting hair growth?
The students made predictions of the rate of hair growth based on a video depicting the hair growth of a young man who grew out his hair for "Locks of Love". Students measured their own hair and researched how long hair must be in order to donate it.
Students then used the data to create a graph and to model it using an algebraic equation. They then gave a presentation on their findings and defend their predictions and mathematical calculations.

Another teacher provided an example PrBL which challenged students to create a new snack using chocolate and peanuts which could not exceed specified conditions regarding weight and fat content. This project required students to apply systems of inequalities to a real life scenario.

A third teacher provided an example PrBL which asked students to use trigonometry to analyze the "biorhythms" of the two quarterbacks in the superbowl to predict which quarterback would win the superbowl.

The teachers provided the following suggestions about implementing PrBLs:
First, student grouping and collaboration is important for driving discussion and investigation. They suggested allowing students some amount of choice in who they work with, so they are enthusiastic about working together. Students must be willing to discuss ideas in order to work together to achieve the expected results.
Second, the teacher should provide students with a rubric so that they have a goal to aim for. Students find it frustrating to work on a project if they are confused about what the teacher desires.
Third, the teacher should ask many questions to guide students' thinking. Instead of telling students "that's wrong", they should challenge the students to defend their ideas and use mistakes as a base for learning.
Finally, authenticity is extremely important. Students have to be "hooked" to get passionate about what they are doing.

PrBLs are excellent for developing 21st century skills, helping students understand how to really use math in their lives. Memorizing mathematical facts and formulas is not enough.

They provided the following site for people interested in PrBLs: www.pbl-online.org

2013 NCTM Conference: "I See It: The Power of Visualization"

2013 NCTM Conference (4/18/13 - 9:30 am - Marc Garneau):
The second session I attended at the 2013 NCTM Conference was Marc Garneau's "I See It: The Power of Visualization". As a visual learner, I know how helpful it can be to provide students with more concrete representation of mathematical concepts.

Marc suggested that most students have been taught symbolically their whole lives, and the lose touch with what the symbols really represent. Students who do not understand what the symbols really mean have no idea how to visualize math or even to apply it in real life applications. Visualizing math can help students reason about math and give them a fuller understanding of the abstract concepts.

A typical example of visualization is creating diagrams to depict fractions. These can be very helpful building students' understanding of fractions, which can be a very challenging concept for students. Comparing fractions can be difficult symbolically, but is much simpler if students are able to depict fractions visually and comparing the concrete representations of these numbers.

A more exciting example of visualization for high school teachers is the use of patterns to develop algebraic reasoning. Teachers can use visual patterns to first help students develop recursive reasoning (i.e. "look at these two first figures in a pattern: how many squares will the third figure have?") , and then to build to a more abstract understanding of algebraic patterns. Teachers can use patterns to prod students to reason about what the 10th or 100th item in the pattern will look like, without recursively counting the items in each step. This helps them understand algebraic modeling without even realizing what they are doing.

These patterns can be very simple or quite complex. Marc showed us a quadratic pattern that can be used to help students begin to explore quadratic functions, which could be a much more tangible introduction to this family of functions.

2013 NCTM Conference: "Communicating Performance for Common Core State Standards"

2013 NCTM Conference (4/18/13 - 8:30 am - Forrest and Elizabeth Clark):
I was first introduced to standards-based grading (SBG) by Shawn Cornally at Think Thank Thunk. I thought the idea was a fascinating one, but I was interested in learning more. When I heard that Forrest and Elizabeth Clark were giving a session on their experiences with SBG, I knew I wanted to come collect the fruits of their experiences.

Forrest pointed out that assessments should predict what students actually know. He suggested that a gradebook should not be used to direct students' behavior, but should be used only as a measurement of academic progress. Therefore, the gradebook should not be used to score effort, participation, attitude, behavior, or, more controversially, group work (except individualized portions) and homework.

In defense of this stance, he noted that he can predict how well his students will perform on state assessment tests, simply by looking at his gradebook. A good grade indicates that a student understands the mathematical concepts explored in his class, while a bad grade suggests that students do not yet understand these concepts.

The first key to SBG is that grading is based on standards instead of a specific test or quiz. These standards are assessed and graded individually. Students' performance on each standard is easily communicated to students and parents, so all parties know what each student needs to work on. This is very useful for creating a plan to help students receive extra help on the specific skills defined by each standard.

Another key component to SBG is allowing students to be re-assessed if they performed poorly. This allows students a second chance to relearn concepts for the natural reward of a better grade.

The process for implementing SBG is as follows:
1. Identify learning targets
2. Align curriculm with the learning targets
3. Create standards-based assessments
4. Create the assessment process
5. Inform parents and students

1. Identify learning targets:
The teacher must identify what students should know, do and understand. These concepts should be drawn from the common core standards, but may need to be simplified as needed. It is easily possible to draw 30 concepts from the standards, but Forrest suggested whittling these down to 15-20 concepts that will be measured in the grade book.

2. Align curriculm with the learning targets:
The teacher needs to identify where in the curriculum each of these concepts will be addressed. What order will he or she teach these concepts? Instruction should be modified to flow with the learning targets, instead of meandering aimlessly.

3. Create assessments:
The teacher must create assessments that identify how well a student understands the learning targets, so that each student can receive an accurate score. The teacher can use existing assessments that already address the targets, edit existing tests, or create new assessments, depending on his or her needs.

4. Create assessment process:
Each teacher must ask himself/herself the following questions:
a) When should assessments be given?
b) When will re-tests be given?
c) What is required of students before a re-test?
d) When will extra help be provided?
Forrest provided plenty of ideas from his own classroom. He requires a mandatory re-test when students score below 80% on an assessment. Students who score 80% or above have the option to re-test as well. He provides the first re-test during normal classroom time (students who are not re-testing work on homework or other assignments during this time), but any other re-tests must be done before or after school. Students are only tested on specific targets that were not mastered, so the re-test may not be as long as the original test. He provides extra help as he can, during his planning time or before or after school as students have time. The re-test score replaces the original score, for better or worse.

5. Inform parents & students:
The teacher should let parents know the grading policy as often as possible, until they are used to the "new" style of assessing students' understanding. Forrest and Elizabeth suggested including the grading policy in the syllabus, in school newsletters, etc., and should explain the process the process to parents during open houses and parent nights.
Parents should be told that grades are determined by assessments only, but good behavior is still important and will be enforced in other ways. Parents need to understand that if a student scores below the standard on a learning target, the student will be re-tested and the most recent score will replace the earlier score.
Finally, it is important for administrators to be on the same page, or the SBG process will not work. If the principal does not have the teacher's back, the revolutionary style to grading will only cause strife.

The Taste of Success

Some of our students have a problem with motivation. Even though they know how important school is, they don't have the drive to succeed. Getting them over this hill has been one of the most significant challenges I've faced.

I've found a solution that seems to work better than anything else I've tried. If I give them a small and very attainable goal, this will give them a chance to succeed - even in the tiniest of ways.

Once they know they can succeed, motivation increases a hundredfold. That taste of success builds their confidence and self-esteem and helps them to believe in themselves. From here, I am able to help them form bigger goals at which to aim. With their newfound confidence, they are much more eager to pursue bigger and better goals as success builds upon success.

There are still moments when they forget the taste of success, and we have to start over again. But once they taste it again, I can see them lift their eyes to the stars, and I smile.

Inquiry-Based Learning

I believe that often learning is more about asking the right questions than providing the right answers.

I am a huge fan of inquiry-based learning, where students learn by seeking out the answers to questions, whether they are prompted by the teacher or the student. With the student population I work among, this usually means that I'm the one asking the questions to propel learning. Rather than telling my students the "answer" when they have a question, I try to ask a question to make them think. If they aren't able to answer my question, I ask another question, trying to build on what they already know.

Sometimes I have to give raw facts or the "right answers". But it is my goal to ask questions first, to help my students think and to challenge them to explore math for themselves. I would a hundred times rather that my students leave my class having learned to think critically than that they leave having memorized a few formulas that they'll forget in a matter of months.