Like many Nebraska high schools, we are trying to improve our students' scores on the NeSA-M, our state's 11th grade standardized math test. We have a 35 minute period used to stagger our lunch on Tuesdays-Fridays in which teachers teach courses for enrichment or remediation. What follows is the description of one such day in my class for juniors designed to help prepare them for the NeSA-M.

I should also note that from a philosophical perspective, I think it's way more important that I teach my children mathematics well as opposed to teaching them just enough to get through a standardized test. Our department has taken the position that we could really care less about the test. Sure, we care if our students do well on the test, but it's way more important for us to focus on the 1,080 hours of instruction students will get in the classroom rather than one snapshot of what they remember on a single day.


We spend some of our time in "NeSA-M CATS class" talking about the formula sheet students get to use on the state test. Our state test does not allow for calculator usage. I have some strong opinions on this, but that's a story for a different day. Because students can't use a calculator, I try to be intentional about pointing out opportunities to leverage the distributive property, factoring, association, subtraction, etc. to make the mental arithmetic a little less problematic. The three red arrows on the screenshot of the formula sheet indicate the computations I wanted students to focus on while we discussed the problems below. Students interacted in small groups and discussed methods of solution. I called students to the front of the room to present their thinking at the Promethean board. Then, we discussed as a class whether we agreed or disagreed with the logic presented by the students. Below is a brief summary of the highlights of each problem.


Problem 1: Given two ordered pairs, find the midpoint

The endpoints of a line segment are (12, 3) and (10,5). What is the midpoint of the segment? 

With my students I emphasize that when we wish to find the midpoint of two ordered pairs, we should split this task into the average of the x's and the average of the y's. I used the metaphor of streets in a city to communicate this point. For example, if you live on 5th street and I live on 11th street, then we should meet at the 8th street cafe for lunch. But if you live on 5th street, and I live on 14th street, then we will have to meet in the middle of a block rather than on a corner for lunch - since the average of 5 and 14 is 9.5. Below is a screenshot from what I put on the board after a student showed her work at the front of the room, work that was simply substitution into the given formula. I simply wanted the students to see beyond the exercise and to understand the concept of finding an average and its application to computing distances.







Problem 2: Find the area of a given trapezoid.


I am bothered when I ask students, "How do you find the area of a trapezoid?" and they reply with something like, "I don't remember the formula." Students spent time working in small groups computing the area; many were successful. I think their success could be attributed to this being a tidy exercise in substitution. We spent time kicking the expression provided by the formula sheet around algebraically. I wanted students to notice the connection to the previous problem, that we can shift our thinking once again to thinking about averages. We can think of the area of a trapezoid as "the average of the bases multiplied by the height." (see screen capture from Promethean board below).



Problem 3: Find the area of a rectangle with information about the rectangle's perimeter.


This is the problem I want to feature in this post. I had students work once again in small groups to solve this problem. Keep in mind many students in the room were in math classes below Algebra 2 (as high school juniors) and many of these students struggle with math and math efficacy. After three minutes working in groups, I asked if any student that hadn't gone up yet wanted to share their solution method. I had a student go to the front and write the following. (It's in my handwriting, because he quickly erased it since he thought he was incorrect).



I experienced tunnel vision myself because I was so used to working problems like this a particular way. At first, the student wrote down the two division problems, as they are shown above, and then simply circled 98 as the answer. With a little coaching, the student wrote the rectangle on the left and labeled its dimensions. I asked if any other student could go to the front of the room to explain the student's thinking. A girl quickly said, "I've got it!" She, too, struggles in math class, but went to the board anyway. She said it was like the averages we did earlier. I still couldn't reconcile the comment and what the students described. After a minute or so of heated discussion, we understood what the boy was trying to say.



The problem was "like the averages" because we can think of three sections of fencing material, each with an average length of 14. In other words, the two lengths of the rectangle form the two segments 14 cm long, but the third segment of 14 is really the sum of the two widths. A formalization of the student's work reveals he chose to use the length as his single variable.

I couldn't immediately understand what he was doing because I was so focused on using the width as the single variable. This approach is really a personal preference, because using the smaller quantity as the single variable helps us potentially avoid fractions. (The constants in this problem are written in a way to facilitate mental arithmetic, but we can't always assume the constants will be so pretty). Another student wrote up on the board the solution method using the width.



Even though the math here isn't tough, I felt very proud of myself as a math teacher during this lesson. Because I chose my words carefully and never confirmed nor disconfirmed whether the student was right or wrong, the students in the classroom responded by unpackaging their thinking. The students had a conversation filled with respectful disagreement and clarifying questions like, "what makes you say that?" This was early in our rotation, so I knew very few of the names of the students in the room. The smiles on the kids' faces were great to see. Just before the bell, a student said, "I wish I could learn math this way all the time."

So, the challenge for me as a math teacher and a leader of teachers is to figure out how to honor the student's comment on a large scale in our school. As the students filed out of the room, I spun in my desk chair, opened my fridge, grabbed a yogurt, and thought about what would need to happen in all math classrooms to make this type of discussion possible.

I keep going back to the idea that the teacher is so critically important. The teacher poses the task. The teacher asks the questions. As teachers, we have ample experience with students asking underdeveloped or vague questions. Low quality questions get low quality answers. On the other hand, high quality, well-formed questions get high quality, well-formed answers as time allows.

So, here's the question: What professional development sessions, teacher collaboration sessions, or activities already exist that help teachers refine their ability to ask high quality questions and cultivate an environment facilitating mathematical discourse?



1 Comment

Many standardized math tests (the ACT, the SAT, and AP exams) allow students to use graphing calculators. The most common model here in the U.S. is the TI-83/84 family. I suspect many test proctors do not know how to check these devices for programs that may be used to cheat on these exams.

No, I have not had to deal with this particular problem... this year.

I have a classroom set of calculators on which I periodically check to make sure there are not any programs or apps that might help students cheat. Every time I administer an assessment, if a student uses their own graphing calculator, I walk around the room and physically check for programs and apps that might help them cheat.

Here is a PDF I made on how to check for cheating on the TI-83/84. This covers programs only. A student could potentially download an app with content disallowed on a test. To look at the apps, a teacher would need to press the APPS key and look for suspicious apps that are not standard installs on the calculator.

Checking a TI Calculator for Cheating

I have also posted images from the PDF below.















The education world says instruction should be 'data driven,' but unfortunately, many administrators and teachers have little experience with conducting an exam autopsy analysis. Below is a copy of the form I made and use to analyze data from assessments for unit exams. I shared this form with my department last year. We used it to analyze midterm and final exam data. Feel free to share this form with others.


Exam Analysis Form

Here are some things I think about when I use the form above.

  • Numerical summary: I type the student's grades sorted in ascending order (without identifiers, of course) in Geogebra. I prefer to make a dotplot and a boxplot to display for students on the Promethean board. We use the quartiles to make verbal statements about performance: "Looks like the first quartile is 71, so 75% of the class scored a 71 or better." I can have powerful conversations with my students about analyzing data given their emotional investment in their own performance. I can ask questions like: "Which is preferable? An exam with a middle of the road mean and low standard deviation, or an exam with a decent mean but high standard deviation?"
  • Brief verbal description of exam format: I prefer a balanced approach between different question types, such as multiple choice, short answer, true/false, matching, and sequencing questions when possible. If I see poor performance for a whole section, I might reflect on whether to change the question format in hopes to capture more information to assist my diagnosis of student misconceptions or misinterpretations.
  • Five most frequently missed test items: To conserve class time, I prefer to spend time answering only the most frequently missed test items. If a student has another question, I encourage them to visit with me outside class. If I see a recurrent theme in the content of missed items, I know I probably need to revisit my lesson plans and unit activities to try and isolate the cause of the misconceptions.
  • Five items most frequently answered correct: We all need to try and catch kids in the act of doing things right. However, if everyone is getting an item correct, I might need to ratchet up the item difficulty or ask a deeper question.
  • Verbal description of content assessed: I have this information for my own reference. I use it to aid my construction of learning guides (some refer to these as curriculum maps or pacing guides).
  • Qualitative data regarding instructional methodology & delivery: I make notes of the instructional techniques I used during the unit. These may include guided lecture, partner work, small group work, cooperative formative assessments, writing tasks, and other instructional strategies. If students struggle mightily on an assessment, I may need to examine my delivery of the unit and make modifications to the activities to increase the probability of student retention.
  • Comments: This is the field I reserve for logistic issues going on at school. Perhaps an unannounced fire drill takes place 20 minutes into the exam. Maybe we are on shortened schedule for a Homecoming assembly. Perhaps the students mentioned having tests in almost all their other classes on that given day. All of this information is useful for the planning and preparation phase of teaching.