This guest post has been contributed by Emily Tietjen. You can contact Emily at email@example.com.
Emily was a student of mine as a statistics major at Cal Poly. She was an invaluable help to me as an exceptional teaching assistant for several years*. I was delighted when Emily decided to pursue a teaching career. She has taught AP Statistics and other math courses at high schools in and near Merced in the central valley of California. Since the beginning of her teaching career, I have very much enjoyed visiting Emily and her students every spring. Emily has quickly moved into an administrative role, as she now serves as one of two math coordinators for the county of Merced. In this role she helps teachers throughout the county to teach mathematics (and statistics!) well. I greatly appreciate Emily’s writing this guest blog post about some questions that she encounters in her position.
* In addition to very helpfully supporting students’ learning, Emily also displayed an indispensable but unteachable quality for a TA: She laughed at my jokes no matter how many times she heard me tell them in different classes over many terms**.
** I’ll be curious to know whether she laughs at this one as she reads for the first time.
The first thing that you should know about me is that I could easily be referred to as a fangirl of both Allan Rossman and Jo Boaler. I had the distinct privilege of sitting through six years worth of Dr. Rossman’s courses as both a student and as his TA. Six years included both a statistics degree and a math credential, but honestly, who doesn’t want to spend as much time as they can in San Luis Obispo?
I can confidently say that I gleaned more from sitting through repeated classes from Dr. Rossman than I ever got from any professional development. Ideas that were intrinsic to his style of teaching, although we never directly discussed his philosophy, are concepts that as a new math coordinator I’m only beginning to have a name for. Ask good questions? I used to think that had more to do with the person asking the question: Were they articulate and educated and thoughtful enough to ask a really good question? What I’ve come to understand is that asking a good question means to give the learner the authority to come to an understanding of a concept through their own intuition.
But asking a good question is intimidating for someone (yes, me) who regularly harbors the feelings of imposter syndrome. In this post I will pose some well-intentioned but ultimately misguided questions about how students, educators, and adults view mathematics and primary and secondary mathematics education. I will also discuss why I consider these well-meaning questions to be problematic.
1. Are you a math person?
Many people ask this question of each other and of children. I have been asked this question often.
I grew up in what you might describe as a humanities family. My mom studied English and German and taught both but primarily German. My dad and brother both majored in history, read voraciously, and after teaching the subject both went into administration. I’m like them. I was a teacher, and I’m now an administrator. But I was also never quite like them. It shows in the directness I expect in an answer given to a question and in the long (interesting, albeit) stories my mom tells before finally getting to the point. It shows in my ability to remember numbers and to quickly solve problems and their ability to remember historical events and the interwoven understanding of how they overlay onto each other. Math basically always came easy to me, and reading basically was always quick and comprehensible to them. Clearly, I’m a math person, right? Wrong. As a child, I enjoyed puzzles. My parents praised my efforts. In school, I liked math and they constantly reinforced my abilities. Despite that, each of my elementary teachers (female, for the record) would talk about their favorite subject, which never included math, while I rolled my eyes at the thought that girls could not be as good at math as boys.
Over the years, thanks to many privileges I had, none more powerful than my parents’ faith in me, I took honors and AP math courses with many inspiring teachers. Even more incredibly, I had two particularly wonderful math teachers who were women for geometry and AP Statistics. Both teachers brought math to life. They made our classes collaborative and relevant to the world around us. In both, I was asked to collect data from the outside world and apply meaning to what I had gathered. They gave me manipulatives and visuals and allowed my classmates and me to formulate our understandings of the math. They provided context that made the math meaningful to me. Most of all I had fun.
On the other hand, in most of my language, literature, and social science classes, teachers overwhelmed me with reading, taught history by having us read chapters aloud from a textbook, each student reading one paragraph at a time, followed by showing movies that partnered with the time period (yes, Mulan was shown with our unit on Chinese history). I had a much more meaningful experience in school with math. And I realize that others have stories like mine but completely in reverse.
The work of Jo Boaler (see her book Mathematical Mindsets and her website here) has brought forth research about how brains learn and grow. Her work demonstrates how there is no research that makes someone have a “math brain.” Additionally, everyone has the capacity to continue learning any subject. A combination of factors led to my positive experiences with math. My parents reinforced my ability. I had teachers who empowered me and my learning. There’s no need for the question of whether or not you’re a math person, because there is no such thing. All students can learn math.
2. What class best meets the needs of the student?
This question is often considered as a student is being placed with a particular teacher or in a particular course. Will it be “grade level” or “honors” or “remedial” or …? This one is so hard for me. We want to do the best thing for our students, right? We want to make sure that students who are exceeding expectations are given enrichment and opportunities to accelerate learning and students who are struggling are provided with support and remediation. That sounds good, right?
I have classified this as a problematic question because even though it sounds innocent, it’s really about a practice called tracking. The problem is that research doesn’t back this up. Ability grouping and tracking lead to differential outcomes for students. At the secondary level, trying to meet the needs of where students are at means that teachers spend barely over a third of the year on grade-level material. When students are given grade-level material they succeed more often than not yet they aren’t given the opportunity most of the time. By tracking a student below grade-level content, a district is ensuring that those student’s will never be able to fill the gap between where they are and the grade-level content they deserve to see. Students can be provided opportunities for advancement without needing to create specialized courses and should demonstrate that they have mastered the material before they advance rather than skipping concepts. (You can read more about tracking issues in the reports here and here and also in the NCTM’s Catalyzing Change book series here.)
Another area where we suffer with this question is our undying race to Calculus in high school. Too often we focus on how to prepare students to study calculus rather than consider what courses and skills would best serve their overall education and potential career. The vast majority of jobs in this country will depend on data literacy or statistics, yet statistical topics are typically found in the last chapter of the books and treated as the content that they’ll get to if they have time which they very rarely do. Many of the above links also discuss the need for statistics and data literacy in the TK-12* educational system as well as the problematic nature of tracking. Understanding data and statistics provides students relevance both to their current lives, through contexts that are inherent to subjects they are studying, and also relevance to their future careers. When I was teaching math, students constantly asked when they would use the subject in their “real lives.” When I was teaching statistics, students never asked that question.
* TK stands for Transitional Kindergarten, a preliminary class to Kindergarten offered to children born in September – December.
Fortunately, there are efforts being made to encourage the prioritizing statistics and data literacy at the TK-12 level. For example, Jo Boaler and her team have released a set of lessons on data science for grades 6-10 (here) along with an online teacher course (here) for data science and 21st Century Teaching and Learning. California university systems have considered adding quantitative reasoning courses as an addition to their subject requirements (see here) for applying to their schools (minimum course requirements to be accepted to public universities in California). High school courses have been designed to address the needs of having a more relevant, equitable math course that highlights the use of data and statistics. School districts and states have restructured their pathways that work to remove the tracking that is prevalent within our educational systems which leads to more equitable outcomes with specific inclusion of a statistics pathway.
This work must continue, as we know that data literacy will be crucial for our society to understand. We face the need to comprehend data in multiple ways as we are constantly facing the collection of our own personal data on a daily basis and mostly have no practical access to knowing the ways in which it is used, for the good and the bad. On top of that, more often than not, the careers our students will go into will require the utilization of data and being able to analyze it.
3. Why do we have to do word problems?
Students often ask this of their math teachers. I’m imagining my former students’ voices as I consider this topic. Heck, I hear my teenage self still wondering this.
Assigning word problems is sure to create anxiety, at least with the typical way that we approach them. However, students often struggle with word problems for the wrong reason. The very prospect of word problems ignites so much fear in students that they are hesitant to even read them in the first place. Speed is all too often valued in the classroom and struggle is not, so confronting a word problem is asking students to work on a concept they’re likely still grappling while adding an additional complicating layer. Anyway, students see it as complicated because of the typical way we present it to them. While the traditional pressures of math still exist in many classrooms, or even worse, at home with little to no support. They’ll need to read, decode, create, and image or model, and transform that into something that they can then solve. I’d argue that we teachers haven’t done a sufficient job of preparing students for these situations. It doesn’t have to feel this way.
For example, we can expose students to a context and help them make sense of it before they even know what the question is. By initially excluding the question, students are relieved of the solution-finding inclination that we all too quickly jump to. One of my favorite routines (see here) encourages students to suggest questions that would be a mathematically reasonable question given the context before presenting them with a question. After students have engaged in the context without the time pressure anticipated by typical math problems, they’re able to intuit what could and should be tested. This process gives meaning and helps us to understand the value of the problem.
When students have had this opportunity, word problems don’t feel so hard. Word problems should pique interest and provide opportunities to make connections to the world around us. They give us a reason to do math in the first place. My assumption is that they feel hard because we feel rushed to solution finding. Students are infrequently challenged to think slowly about a problem. The pace of the class is often at the speed by which the first correct answer is given. Word problems can instill fear and yet I think they’re truly key to making math feel relevant for our students as long as they aren’t arbitrary for the grade level.
For an example of what this might look like, consider the following background information from free-response question #3 on the 2018 AP Statistics exam (here): Approximately 3.5 percent of all children born in a certain region are from multiple births (that is, twins, triplets, etc.). Of the children born in the region who are from multiple births, 22 percent are left-handed. Of the children born in the region who are from single births, 11 percent are left-handed.
At this point, a class might have a conversation about clarifications they may need for accessing the language used or understanding the context. Then, the teacher could ask students to come up with a question for the context. Depending on age (or maturity level), students may ask questions like, “Where do they live?” or “How old are the kids?” Those questions need to be redirected, because we are looking for mathematical questions. For this context, students may ask, “What is the probability that a student born in the region is right-handed?” This isn’t the ultimate question asked of students on the AP exam, but having students consider their own questions engages them in the context and gives them ownership of the question. A class of students will often come up with the intended question after only a few suggestions*. Pausing to consider other questions will also be helpful to give students insight into other aspects that may be important for solving the problem. These aspects include what types of variables are present, how the information may be organized or depicted graphically, and what given information may be useful in determining the solution.
* The first part of this particular AP question asked: What is the probability that a randomly selected child born in the region is left-handed?
4. What does good teaching sound and look like?
Okay this isn’t technically a bad question. Teachers and administrators ponder this year after and it continues throughout the career of everyone involved. It’s in consideration when hiring, when deciding if a teacher should receive permanent status, and as the years pass and the field evolves and we learn more about equity and what methods work best.
The problem is that it’s very common for people to think of good teaching like how Trunchbull, from the film adaptation of Matilda, thought of an ideal school as “one in which there are no children at all.” Sadly, many teachers and administrators still consider a well-run class to be filled with students who are silent, only speaking when spoken to, and with students who sit down and stay there almost as if they don’t get to exist there as a person.
We should instead nurture classrooms where students are given the authority to take ownership of their learning because students’ learning is more important than the teaching of lessons. Teachers should be talking no more than half of the time. Students should be talking. Mostly to each other. They should be positioned in a way where collaboration is convenient and encouraged.
In my current role, I support mathematics teaching and learning for school districts within Merced county. My office serves students from twenty school districts as well as our internal programs. This accounts for about 60,000 students, of whom more than three-quarters are eligible for free and reduced lunch. Relative to the state, we have high populations of poverty and students who are classified as English learners. My office uses the following framework, developed by my colleague Duane Habecker based on Maslow’s hierachy of needs, to advocate for an effective mathematics program for all students:
- Material Needs: Every student has a teacher with appropriate mathematics content knowledge and knowledge for teaching mathematics. Math lessons are rooted in a solid understanding of the standards through rigorous, high-quality curriculum and meaningful tools.
- Mindset & Culture: Every student is immersed in a mindset and culture that intentionally communicates all students can learn math at high levels while being responsive culturally and personally in a learning environment that considers each and every student’s unique background, experiences, cultural perspectives, traditions, and knowledge. Mistakes in mathematics are normalized. Students regularly experience high-quality, grade-appropriate lessons and assignments.
- Student-Centered Instruction: Every student regularly experiences instruction that is student-centered and designed to maximize students’ use of language. Lessons create space for students to participate in discourse to promote conceptual understanding, which then leads to procedural fluency, problem-solving, and application.
- Equitable Assessment: Every student is regularly and humanely assessed in order to understand their own growth and to receive productive feedback for next steps in learning. Students use the feedback to know where they are in their learning, assess any misconceptions that need to be addressed, and then use the results to drive the next level of learning.
I hope that these well-meaning but misguided questions have illustrated the misdirected focus that many have about how best to support our students in their mathematics education. When we pigeonhole students into our own fixed beliefs, it’s no wonder that we consistently turn out students who underperform in mathematics as compared with other countries. I believe we will see incredible growth by making mathematics more relevant to students at all ages, discontinuing the use of ability grouping and tracking, and offering more equitable pathways for college and career readiness. Focusing on statistics and data science is a necessary and important part of the solution, as this leads to productive and supportive classroom environments and helps students to acquire essential skills for a modern workplace and world.