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December 1, 2012
Vol. 70
No. 4

Instigating Thinking in Math Class

To help students master the Common Core Standards for Mathematical Practice, teachers must engage students in productive struggle.

Write algebraic equations? Check! Counting principle? Check! I've spent a good portion of my six years as a middle school math teacher checking boxes on a list of skills students needed to master.
Then Maryland adopted the Common Core State Standards and I participated in a yearlong training to prepare teachers to effectively implement the standards. Through this training, and experimenting with how I teach, I've identified ways the Common Core math standards differ from current standards—and ways teachers' practice must change. I'm already seeing such changes make a difference with my students, more than 25 percent of whom qualified for special education services last year and more than 20 percent of whom were English language learners.
The most significant difference is how my role as a teacher changes. To successfully implement Common Core standards, a classroom needs to become student driven and student led. The teacher becomes more of a facilitator, an instigator of thought, encouraging students to persevere through the process of finding a technique for solving new problems.
The second difference is that my checklist of skills is gone. There are seemingly only five Common Core units applicable to middle school math (Number Systems, Expressions and Equations, Functions, Geometry, and Statistics and Probability). Although these units mirror skills currently required of 8th graders, the approach taken to developing these skills is completely different. The standards move away from the large quantity of material students were previously exposed to at each grade level and instead emphasize the depth at which skills are developed. Words like formulize and reason replace solve and explain in instructions to students.
This difference is most apparent in the Common Core's Standards of Mathematical Practice. Unlike Maryland's current curriculum, these Standards of Mathematical Practice (or practice standards) emphasize the "how" when it comes to teaching, not simply the "what." They list eight processes and proficiencies that enable students to cultivate expertise in math concepts and to develop problem-solving skills germane to their grade level (see ".")

Productive Struggle

During my professional development on implementing the Common Core standards, I examined how engaging students in productive struggle helps them get comfortable with the practice standards. I realized what I needed to add to my classroom practice to ensure that all eight practice standards were brought to life. To allow students the opportunity to persevere, use tools appropriately, and so on, I needed to release control of my classroom. I needed to let students struggle—independently and collaboratively—using inquiry as the primary instructional tool.
The established lesson practice within Baltimore City Public Schools is "I do, we do, you do." Teachers introduce a skill; students then complete examples as a class, with the teacher guiding them through each step; then students demonstrate their learning through independent practice.
Inquiry-based lessons, in contrast, begin with "you do" and give students the opportunity to figure things out for themselves. They rely on the first of the eight practice standards: Make sense of problems and persevere in solving them. The theory is that if a student works through the steps on his or her own, that student has a higher likelihood of retaining the information than if he or she were simply given steps to follow.
In theory, this approach sounds wonderful. But how can we expect students operating below grade level to succeed if left to their own devices? Given the wide diversity of economic background and proficiency levels in my class, how were my students going to learn to make sense of problems and persevere in solving them?
These questions ran through my mind during my training. Fortunately, I learned approaches that strengthened my ability to teach through inquiry, including three powerful takeaways:
  • Teachers must engage students with new concepts—physically and in other learning modalities.
  • Effective questioning can spur inquiry.
  • It's possible to teach math effectively without using directives or modeling.

Teaching with All Modalities

If you physically engage a student while stimulating his or her brain, you create the best chance for that kid to actively participate in inquiry and retain learning. Kinesthetic movement is one of the four modalities (along with visual, auditory, and tactile) that educator Neil Fleming has identified through which people process new information and experiences.
At the beginning of math classes, I do quick warm-ups involving movement, such as "Shake Down, Count Down." Students start at a given number—say, eight—and shake their right hand eight times, then their left hand, their right foot, and their left foot. Once you've worked out all four limbs, shake each limb seven times, then six, and down to one. In addition to introducing movement, the silliness helps build a classroom culture in which students aren't afraid to poke fun at one another or make mistakes.
I try to keep this inquiry-promoting atmosphere going as I introduce concepts. For example, in a lesson on unit rates, my students first count how many jumping jacks they can do in three minutes, then calculate how many they could do in one minute. I have them browse store circulars to find prices for items, calculate the unit rate of each item they identify, and complete a worksheet showing those calculations.
We then discuss the items and prices students chose and how they calculated the unit rate of various items with the information they had. Class discussion not only uses auditory skills, but also enables students to engage in several of the standards for math practice, such as critiquing others' reasoning.

Questioning Effectively

As an instigator of thought, I try to ask the right question—one that enables unsure teenagers to progress to the next level of learning. I've learned to use "hook questions" to engage students' curiosity and encourage inquiry in math.
A hook question has no one right answer. It doesn't even have to be something to which you know the answer. A good hook question encourages students to become investigators and to seek the answer from outside resources—textbooks, the Internet, even their communities. This open-endedness might also lead students to use several of the Common Core's practice standards, such as, make sense of problems (Standard 1) or look for and make use of structure (Standard 7).
For example, as students learn how to solve inequalities, we tell them that if you multiply or divide by a negative number, you have to flip the sign of the inequality. As I explained this rule, one of my students asked, "Why?" As I thought about it, I realized I wasn't sure. I could show this rule with an example, but I couldn't explain why it existed.
I made discovering the answer to this question students' homework for the night and challenged them to find the answer before me. Students were allowed to seek the answer from any source. They loved the challenge. There was an entire thread about this homework on Facebook as students asked friends in high school to help explain the rule. Even a math teacher from another state chimed in.
I'm confident that most of the students involved will never again forget to flip the inequality symbol. If they do, they can go back to their archived Facebook messages and rediscover why they need to—because when you multiply or divide by a negative number, you change the value into its opposite (a negative becomes a positive and vice versa). To accommodate this change in sign, you must also change the inequality symbol.
My students who struggle with math take one look at a word problem and immediately raise their hand and ask for help. Instead of asking what they need help with, I now ask them what they already know. For example, in a problem involving a system of equations, most of my students know there are four methods of solving: elimination, substitution, graphing, and guess and check. Most students can usually identify these methods, so I ask, "Which method can you use?" Then I allow the student to guess and check their way through the problem or try any of the other methods. At least this gives them a way to plunge in.
A more advanced student may solve the same problem by immediately choosing the simplest method. I'd ask this learner, "How can you answer this problem using one or two methods that differ from the one you already used?"

Teaching a Required Skill Through Inquiry

Let's look at how I taught one skill my students must master with my old approach and then with my Common Core–friendly approach. In Maryland, 8th grade students must be able to write inequalities to represent relationships, using a variable; appropriate relational symbols (>, ≥, <, ≤, and =); no more than three operational symbols; and rational numbers from -1,000 to 1,000. The skill is taught as part of a unit involving writing, solving, and graphing solutions to equations and inequalities.
In the past, I've taught this unit according to the district's curriculum guide, which acknowledges the importance of real-world connections. For example, the problem the district's curriculum guide used to introduce students to writing inequalities deals with wanting to see a PG-13 movie. Students come up with a list of ages that would be allowed to attend, then think of real-world situations and questions for which multiple answers could be correct (such as, "At what ages are people allowed to drive cars?"). The difference between these practices and inquiry-based practices comes in the next step.
In my district's curriculum, teachers next guide students through an activity in which they explore an inequality word problem, such as,
To receive a discount at the Half-a-Dozen Flags Amusement Park, a group of visitors must purchase a minimum of $600 worth of full-day and half-day tickets. Twenty-six tourists in a group purchased full-day tickets at $15 each. Write and solve an inequality to calculate how many more tourists in this group would need to each buy a $9.50 half-day ticket so this group qualifies for the discount.
The teacher directs students to take three steps:
  1. identify the different events,
  2. label them with a variable,and
  3. fill in the values for each event.
In the inquiry-based instruction that undergirds the Common Core State Standards, a teacher would instead ask open-ended questions, as a facilitator might.
In my newly designed lesson on inequalities, I gave pairs of students a "deconstructed" question. This meant I rewrote questions found in a math textbook to include the bare minimum of information and allowed students to determine their own procedures. One textbook question read, "To go to Disney World costs $50 for each adult ticket and $30 for each child's tickets. X tickets are bought of each. If the total spent is more than $800, how many tickets were bought?"
I deconstructed this problem so it would push students to demonstrate Practice Standard 1 (make sense of the problem and persevere in solving it) without frustrating them with extra words and exact figures right off. (This adjustment especially helped my English language learners and students with individualized education programs). I gave them this question: "You're taking a trip to Disney World. How many tickets do you buy?"
Initially, students clamored about not having enough information to move forward. Once they were given the instruction of solving the problem any way they wanted as long as they could explain the process they followed to get their answer, they were on their way. Some students were practical: They bought enough tickets for their family and friends. Others bought enough tickets for the Baltimore Ravens football team or all members of their favorite boy band.
After students had presented their ideas and we discussed them as a class, I asked, "How much money would you spend to take all these people to Disney World?"
This time, students were energized and focused on the process of discovery. One group used the Internet to figure out the actual cost of a ticket. Another said they found a coupon for a 100 percent discount so didn't have to pay anything! By the time we were finished, all my students could identify what key information they would need to solve a word problem like this and what simple operations they could use. When I presented them with the original textbook question, they could make use of the structure they'd discovered and see this complicated question as a set of simple steps.
  1. Multiply the cost of an adult ticket by how many adults went.
  2. Multiple the cost of a child ticket by how many children went.
  3. Add for your total.
  4. Make sure this sum is larger than $800.

Investigating Results

As I continued changing my teaching approach throughout the 2011–12 school year, I wondered if this transition from knowledge provider to instigator of thought was benefitting students. I decided to investigate by analyzing test results. My school district requires students to take an assessment in February that mimics the end-of-year Maryland State Assessment and covers all the math skills 8th grade students are accountable for.
Overall, the performance of my students on this benchmark assessment was poor. Only 17 percent scored in the proficient or advanced range, meaning their score was at 60 percent or above. Of the remaining students, only 20 percent scored in the 50 percent range. These results seem to suggest that my inquiry-based instruction approach aligned to the Common Core's demands wasn't working.
However, an item analysis of each question told a different story. By this February benchmark, I'd only taught four inquiry-based lessons. On average, students scored better on three of the four test items that assessed skills I'd taught through inquiry than they did on items assessing skills I'd taught traditionally. For example, 75 percent of my students could recall my inquiry-based lesson on the skill of writing inequalities and selected the correct answer.
In the next few years, math teachers will discover that their roles must change if they are to help students meet these Standards for Mathematical Practice. After spending a year exploring these standards, I have three general suggestions for how to start this shift.
First, take time to read and understand the standards for mathematical practice. Identify which ones you already emphasize in your classroom and analyze how you must change your classroom so that you can strengthen students' skill with the remaining standards.
Second, join a professional learning community, which will provide you with the support necessary to make difficult transitions, commiserate about failures, and celebrate successes. If there are no opportunities through your district, take advantage of national conferences or webinars on the Common Core standards available through many associations.
Finally, realize that you have the power to create change for your students. So much of education reform is done to teachers and students. This is a change that can be done by teachers. By equipping yourself with the tools to engage, challenge, and enrich students' learning experiences, you can transform schools into laboratories in which students learn to successfully investigate, discover, and solve problems.

The Eight Standards for Mathematical Practice under Common Core

  1. Make sense of problems and persevere in solving them.

  2. Reason abstractly and quantitatively.

  3. Construct viable arguments and critique the reasoning of others.

  4. Model with mathematics.

  5. Use appropriate tools strategically.

  6. Attend to precision.

  7. Look for and make use of structure.

  8. Look for and express regularity in repeated reasoning.

End Notes

1 Fleming, N. (1995). I'm different; not dumb: Modes of presentation (V.A.R.K.) in the tertiary classroom. Proceedings of the 1995 Annual Conference of the Higher Education and Research Development Society of Australasia, 18, 308–313.

2 Fried, R. L. (2001). The passionate teacher: A practical guide. Boston: Beacon Press.

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