My district recently completed the first quarter of the school year. I think this is a great time for teachers to reflect on their first quarter and think about how they can improve instruction to increase student achievement for the remainder of the school year.
We often hear about self-reflection and many teachers understand the importance of this concept. However, I’m not sure how many teachers know how to best reflect on their teaching practices. Below are a few strategies I have used in the past. Many of the ideas came from an NCTM article I read years back, and I’ve tried to systematically implement these ideas into my routines. Not every strategy works for every teacher, but it can be helpful to review some of these and then develop some of your own:
- Think about student learning that has occurred. What has been successful? What content have students missed? How are you going to cycle back and review content that you aren't confident your students have grasped?
- Ask students to reflect on their own work during the marking period. This is more appropriate for older students. Ask questions like: What was most challenging for you? What topic did you find most interesting? Of what math accomplishment are you proudest? This can help students understand how the math they learned holds meaning for them.
- Think about your best and worst lesson over the past several months. What made the lesson successful or unsuccessful? Do you intend to use the lesson again next year and, if so, what changes need to be made? Reflecting on necessary changes and jotting down notes is actually a great strategy to use after each lesson…it makes planning for the subsequent year much more manageable.
- Talk to colleagues about your successes and challenges over the past few months. This is especially effective early in the school year when you are still working out the kinks. Collaborate with others about how to tweak instruction over the next several months to ensure greater student success.
With our busy schedules, taking the time to deliberately reflect, write down notes, and strategize to make changes can seem overwhelming. However, I’ve found that taking just 30 minutes at the end of a marking period (or more often, if possible) can really benefit my teaching practice. What other types of self-reflection have helped you improve your instruction?
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Report after report shows that the U.S. ranks near the bottom in student math achievement among other countries, but what if we broke up the rankings by state? If we treated each state as its own country, how would they fare among the best in the world?
A recent article in Atlantic Magazine
tackled that very question, highlighting the work of Stanford economist Eric Hanushek and two of his colleagues. Their findings: "Even if we treat each state as its own country, not a single one makes it into the top dozen contenders on the list. The best performer is Massachusetts, ringing in at No. 17. Minnesota also makes it into the upper-middle tier, followed by Vermont, New Jersey, and Washington. And down it goes from there, all the way to Mississippi, whose students—by this measure at least—might as well be attending school in Thailand or Serbia."
Massachusetts has not always ranked at the top of the states, and several changes are thought to have given them this boost:
- The state requires students to pass a test before graduating from high school.
- To help tutor the kids who failed, the state moved money around to the places where it was needed most.
- Massachusetts made it harder to become a teacher, requiring newcomers to pass a basic literacy test before entering the classroom. In the first year, more than a third of the new teachers failed the test.
Those improvements, however, are seemingly not enough for to make our best state competitive among the rest of the world. The U.S. Math Performance in Global Perspective report released this month challenges many common beliefs about what our education system needs. Perhaps our focus is in the wrong areas. Though controversial, research has suggested that class size and the amount of money spent on students are not related to math achievement. Others have argued that having a population that is heterogeneous is difficult to educate. However, the study by Hanushek and his colleagues shows that on standardized math tests, only 8 percent of white students in the U.S. class of 2009 scored at the advanced level, a percentage that was less than the share of advanced students in 24 other countries regardless of their ethnic background.
It's a big question, but what do you think is the reason for our country's not-so-great math performance?
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I’ve been preparing some in-school professional development for teachers in my district that requires me to look at best practices for mathematics classrooms. Much of our recent attention has been on the new standards and updating curriculum, but I wanted to find some practical advice that teachers can use in the classroom immediately. I came across an older article in Teaching Children Mathematics
that provides some practical suggestions about ways to strengthen math in your classroom.
- Create a mathematics environment – Provide students with daily reminders of how math is used outside the classroom. Allow students to explore different areas of math independently.
- Make mathematics a priority within your classroom – Integrate math with other subject areas. Make sure students know that math isn’t just used during “math time”.
- Plan to connect with parents – Be sure parents know what math their students are learning. Provide them with opportunities to explore the math with their students.
- Take an inventory of your mathematics materials – Sort through materials that you don’t use often and think about how to integrate them in to lessons. Then, make a wish-list of manipulatives that you may find useful during upcoming lessons.
- Seek opportunities for professional growth – Even if professional development programs aren’t available, look for books and other reference materials that can keep you up-to-date on trends in math education. (As a reader of the Math Hub blog, you’ve already got a leg up on this one!)
- Make problem posing an integral part of your mathematics curriculum – Engage students’ curiosity and encourage them to ask questions and create their own mathematical problems.
- Work with your parent organization – Plan a school-based activity that will get parents and students investigating math together.
- Share your success – Create a learning community by sharing great ideas and lessons with other teachers. You’re likely to learn as much as you teach to others!
- Take a leadership role in mathematics – Help novice and experienced teachers grow by facilitating discussions involving mathematical topics and lessons.
- Become an advocate for mathematics – Share positive experiences with others in the community (board members, newspapers, etc.) Don’t keep your school’s successes internal…be sure everyone knows about the great work you are doing.
As professional educators, we are often receiving advice about how to better our teaching practices. Sometimes it’s the simple suggestions like the ones above that are just what we need to spark positive changes in the classroom!
Recently, an article was published that details the findings of a study comparing the mathematics achievement of students in the United States to the achievement of students in other countries. Much of the information was not surprising; we are continuously learning of studies that show that U.S. students are falling behind other countries in math and science achievement. One facet of the article, however, got my attention. With so much focus on NCLB and ensuring that our struggling students “pass the test”, what are we doing to advance the achievement of students who are already deemed proficient or advanced based on their past achievement? How are we helping these students move forward along their educational path?
At my school, gifted students are provided with enrichment activities and offered some opportunities to advance their learning beyond the classroom setting. In previous years we had a Gifted and Talented program, but with recent budget cuts, that program had to be sacrificed. Now, it is up to the classroom teachers to provide enrichment activities for these students during free-time in class or after school. With the concerns teachers have about helping their struggling students, I can’t help but think that many gifted students are not provided with opportunities to excel.
My assumption is that the recent down-play of enrichment opportunities in my school is not uncommon. Instead, I believe that there are talented students throughout the country who are not provided with opportunities to advance their talents. There is so much focus on ensuring that all students are working on grade level, but I think some students who are able to work above grade level are slipping through the cracks. I think that this is an area that needs more attention if we are meant to compete globally in math and science. We don’t just want U.S. students to be average; we need them to be exceptional! How are teachers in your school helping students become exceptional? I think we all could use some ideas about how to make this work!
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Last month I attended the Conference Board of the Mathematical Sciences Forum (CBMS) on Content-Based Professional Development for Teachers of Mathematics in Reston, VA. The CBMS is an umbrella organization representing seventeen professional societies, including math education groups like NCTM, NCSM, and TODOS; associations of mathematicians, such as the Mathematical Association of America and the American Mathematical Society; and other math-related groups. This is the third CBMS forum I've attended. Two years ago the focus was the Report of the National Math Panel. Last year we concentrated on the emerging draft of the Common Core State Standards. The goal of each forum has been to generate discussion and distill policy recommendations for decision-makers at the federal level. With a focus on teacher professional development, last month's forum tackled what is arguably the most important lever for improving math performance. The speaker and panel presentations are available on the CBMS website. I just want to share a few quick highlights:
- Several speakers used different terms – structures, trajectories, content progressions – to describe the same idea. The Common Core State Standards is built on content and skills that connect from year to year. The objectives should not be considered in isolation. They are part of threads that run, for instance, from Base Ten Number and Operations through the Number System and into Algebra. Understanding those interconnections is critical for teachers, and one of my breakout groups recommended incorporating the trajectories into math teacher education. Elementary teachers need to see where the content they're teaching will be leading their students. And middle and high school teachers need to follow the threads back to what their students learned in 3rd or 4th grade.
- Teachers are learners too. Kind of a "duh" realization, but we often neglect to apply basic learning theory when we're working with adult teachers. We need to respect background knowledge, provide appropriate scaffolds and supports, and connect procedures to meaning. What we do to ensure successful instruction of children, we should also do for teachers in pre-service and professional development.
- We need to provide teachers with new ways to assess student work, particularly student understanding. Much of what currently happens in the classroom focuses on measuring skill and procedural knowledge. What kinds of tasks can provide windows into student thinking and strategic competence? There's much work to do here, but some of the presentations from the forum offer a glimpse of the possibilities.
A synthesis of ideas and recommendations from the forum should be available soon, so check back on the CBMS website next month.
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I want to share an activity that helps students understand the importance of community involvement but also helps them practice important math concepts. I used this activity in the weeks leading up to Thanksgiving, but it is one that can be meaningful at any time of the year. Many schools hold can drives to collect food for needy families. However, I decided to put a different spin on my can drive by including daily math activities to help get students more involved.
Students were asked to bring in non-perishable canned goods over a 2-week period. While participation was not mandatory, I found that most of my students were eager to donate. Each day, my class spent time counting the number of cans that had been collected and graphing the data. Students decided to create three different types of graphs: a line graph, a stacked bar graph, and a line plot. Each graph displayed the information in a different manner and we used the graphs to discuss appropriate data displays. I also had students estimate the weight of particular cans (I had to “hide” the weight on the label) and then they weighed the cans to check the accuracy of their estimates. At the end of the project, we also found the total weight of all cans that were collected! Next, I had students find the volume of some of the cans, and I had them find the difference between the volumes of the largest collected can and the smallest. Finding the volume of a cylinder was a new concept for my students, so it required a bit of guidance, but they loved using the rulers to measure the cans and it required them to measure precisely.
Some other ideas I had but did not use included students looking at the labels to investigate surface area, converting between measurements, and finding the unit price of some of the items. The activities that you can choose are dependent on the level of your students. Overall, though, I found that students enjoyed this because it got them more involved and interested in the charity activity. Typically, students just drop a can off and forget about it. With this activity, they were excited to create graphs, measure the “new” largest can each day, and they were helping their community at the same time!
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Most educators recognize the tremendous influence that parents have on their children’s education. From my experience, however, few parents have a strong understanding of the ways in which students’ perception of math is shaped by the ideas of their elders. I can’t tell you how many times I’ve heard parents say something like “I don’t expect my child to be good at math. I never liked it and I’m not good at it either.” The parents don’t mean any harm, of course. However, it pains me to know how comments like this affect young children. Kids emulate their parents; they adopt their parents’ beliefs; they often want to be just like their parents. So, imagine the ideas that students develop about math when their parents don’t have respect for the power of mathematics.
Recently, a group of researchers completed a study aimed at examining the mathematical interactions between children and parents. According to the study, child/parent communication involving numbers is likely have a positive influence on later student achievement. Studies such as this one demonstrate a direct correlation between parent/child interaction and student achievement. Previous research has shown how early math achievement is a strong indicator of future success, but this study indicates how critical a parent’s role is in this early achievement. It is important that all students enter school with a strong mathematical foundation and disposition. As their children’s first teachers, parents are given this tremendous responsibility.
I don’t fault parents for negatively influencing their students’ perception of math; I just don’t think they recognize the educational impact they have on their children. To many, education officially starts on the first day of Kindergarten. As teachers, we know this isn’t the case. Therefore, we must help parents understand the importance of the foundation they set, as well as the need for them to communicate positively about the importance of math education.
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Not-so-surprising news from the research files: The more anxious elementary school female teachers are about math, the lower the math achievement is for the girls in their class. But here's an interesting twist: The female teachers' anxiety levels did not relate to the boys' math achievement, only the girls'.
Researchers at the University of Chicago published the findings earlier this year. Math anxiety was measured by a 25-item questionnaire that asked teachers how anxious different situations would make them feel (e.g., “reading a cash register receipt after you buy something,” “studying for a math test”) on a scale from 1 (low anxiety) to 5 (high anxiety). When taught by female teachers with higher math anxiety, the girls not only were likely to have lower math achievement, but they were also more likely to endorse the stereotype that "boys are good at math, and girls are good at reading." For boys? No effects on their endorsement of the stereotype.
So, why these different effects on girls and boys? The report explains, "Children are more likely to emulate the behavior and attitudes of same-gender instead of opposite-gender adults. Because early elementary school teachers in the United States are almost exclusively female (>90%; 91% across elementary school and even higher at early elementary levels) and gender is a highly salient feature to children at the early elementary school age, girls may be more likely than boys to notice their teacher’s negativities and fears about math. This, in turn, may have a negative impact on girls’ math achievement."
What do you think: Given these findings, should we rethink the minimal math requirements for elementary education degrees? Or are there other things that we can do to reduce teachers' math anxiety?
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During a recent curriculum meeting, I noticed some colleagues talking and texting during teacher presentations. When I casually made a comment after the meeting, one responded with, “Oh, I didn’t think I had to listen. I don’t teach that grade.” The exchange made me think…why is it so important for teachers to have knowledge of what is taught in other grade levels? Just like many other facets of life, it’s helpful to understand where one has come from and where one is going. Teachers need to understand students’ mathematical foundation so they know how to continue a coherent curriculum. Hence, they need to have a working knowledge of the standards that are required in previous grades.
Similarly, it is important for teachers to understand where students will take their knowledge. Understandably, teachers cannot be expected to know every piece of math content that students will ever see. However, it is important that they have a general understanding about how the content they teach lays the foundation for subsequent math study. For example, I am acutely aware of how critical it is for elementary students to have a deep conceptual understanding of fractions. And, when middle-school students consistently convert fractions to decimals before computing, I try to convince them that, most of the time, working with fractions is much easier. My goal is to eliminate the fear of fractions. Why is this so important to me? Because, previously, I spent three years teaching Pre-Calculus, a course that is reliant on a deep understanding of fractions and fractional computation. This is just one example, but I know that my understanding of where students are going in their study of fractions has helped shape my expectations of my current students.
So, while I know teachers don’t have time to study every facet of math education, it is important to gain a working knowledge of students’ educational paths. Listen to colleagues when they talk about the content they are teaching and share the content being taught in your classroom. You’d be surprised how much this information can come in handy.
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I’ve mentioned that my district is in the middle of re-vamping our math curriculum. The task involves collaboration with several teachers and administrators. With so many different ideas and opinions shaping the curriculum, there are bound to be some issues of debate. One of the major points of discussion is in regards to the use of the textbook in the classroom. As a high school math teacher, I only used the textbook as a tool. I referred to it for ideas, but my teaching was guided by the curriculum and ideas that I gathered from a variety of sources. When I began teaching 5th grade last year, I continued with the same strategies. To me, this made learning more exciting and motivating for the students because I could create my lessons around student interest and ability.
The reasons that I prefer not to rely on a textbook include:
- Textbooks present content in one or two ways and don’t always allow for individualized instruction.
- Many textbooks tend to include low-level questioning techniques and do not require students to think deeply about the math.
- Textbooks do not take into account student interests, strengths, and weaknesses.
- Textbooks often don’t make connections that can be made with intricate well-designed lessons.
Don’t get me wrong, I think textbooks are still necessary in schools. They are a tool for guiding teachers and students, offer substantial practice opportunities, and are a great reference tool when students have questions about a specific topic they may have studied in the past. However, I think as educators, we have to look beyond the textbook and incorporate standards-based teaching that encourages full coverage of the standards and meeting individual student needs.
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I admit...I’ve said it to my colleagues, “That kid is so lazy!” I’m guilty of looking at some of my former students and wondering how they could be so lazy, or how they could lose or simply forget about an assignment. Then, a few weeks ago, I came across an article explaining that many “lazy” students are actually lacking executive skills – the ability to organize materials and break down assignments. While not all students’ apparent lack of effort can be attributed to this issue, many students may be lacking the executive skills necessary to succeed.
The article states that students in elementary schools tend not to feel the effects of lacking these skills because they often have greater support from teachers and parents. As students enter middle school, the level of adult support is weaned and students who have not acquired executive skills begin to suffer. The author uses the RTI 3-tier model to provide suggestions for how to help students in this area:
- Tier 1 (whole-class) Interventions
- Establish classroom routines and rules and review them frequently.
- Develop a method for communicating information to parents.
- Integrate study-skills lessons.
- Plan fun activities as incentives for students to meet certain goals.
- Tier 2 (small-group) Interventions
- Break tasks into small parts – give students suggestions about where to start.
- Establish after-school homework clubs.
- Provide progress reports to parents and involve parents in incentive plans.
- Initiate small-group coaching sessions to teach students how to manage assignments.
- Allow students to use free-time to complete assignments.
- Tier 3 (individual) Interventions
- Define target behavior and criteria for success.
- Establish environmental modifications such as a quiet, distraction-free space to do work.
- Explicitly teach executive skills.
- Provide visual reminders of expectations.
- Monitor student’s use of independent time.
These suggestions aren’t likely to be the magic solution for getting all students to do their work. However, it might be helpful to consider which of your students have the ability but may lack the executive skills to be successful. Then, try some of these tips and let us know how they work for your students!
Dawson, P. (2010). Lazy – or Not? Educational Leadership, volume 68 (Issue 2). 35 – 38.
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I've had quite a couple of weeks of unexpected conversations. First, my wife and I took our son and a couple of his college classmates out to dinner. We've found that food provides a consistent vehicle to maintain some communication with our offspring. Otherwise, we might never see him. One of his friends, Nick, shared his experiences tutoring math in a South African township school over the summer. Nick's job was to tutor algebra to 11th graders, but he quickly found the needs ran much deeper. For instance, Nick was surprised to uncover that the students didn't understand that fractions described equal-sized pieces. The students lacked the most fundamental concept of what fractions mean, yet they were computing away and struggling their way through algebra. I, sadly, wasn't surprised. We see this lack of foundational understanding all the time right here in the United States.
However, I wasn't fully aware of how far students could carry doing without understanding until another discussion erupted in the software design class I teach at the Harvard Graduate School of Education. This year, as in many past years, I have a few students from the Teacher Education Program, folks who are transitioning from math-related careers into teaching. These students are working on a project to leverage technology to help students understand the base 10 system and number systems in general. In sharing the project's progress with the class, one of the students projected a chart showing different bases, from base 2 to base 10, with numbers displayed in the different bases. Some spaces in the chart were left blank, and he prompted the members of the design class to use the patterns to fill a few in.
About two-thirds of the students responded with curious stares. "What's a base 10 system?" one of
the students finally asked. I hadn't expected the question from these Harvard masters students, but the transitioning math teachers and I did our best to explain. I offered the analogy of a car's odometer. When one of the digits in the odometer passes 9 it rolls over to a 0, and the next place value increases by 1. An odometer displaying in a base 3 system would roll over after 2. There's a system, we explained, with 10 symbols in base 10 (0 through 9) for writing any number. In base 3 there are only 3 symbols (0 through 2) for writing any number. Our explanation elicited quiet murmurs of "wow" and "cool". Many of my students had never realized there was a system underlying the numbers they encountered and used everyday. That lack of understanding, however, hadn't prevented them from excelling in math and surviving quite well in the world. They could do, follow the directions and the procedures; they just didn't understand why it worked.
Another student inquiry pushed us to consider whether or not it mattered. "Why would anyone need to learn this?" Good question, one that prompted a more fundamental one: "Why do we teach math?" I just couldn't pass up the opportunity to engage this thoughtful, diverse group in muddling around at least a little bit in this topic. Ever the teacher, I answered the challenge with one of my own. "Have you ever asked this question about literature?" "No." "Great, then let me put it out there: Why do we teach literature?" A short period of quiet thought preceded some suggestions, one being: "We learn literature to understand different ways to communicate." Okay, we don't have students read and analyze Tuck Everlasting, Hamlet, or other literary works because we may use them at a later date. The books are vehicles to learn something beyond the works themselves. We learn about how people think, feel, and express themselves. Cool.
So what about math? The answer wasn't so apparent. We haven't done a good job of identifying what mathematical thinking is and why it's valuable. Math has been something we just learn to do. Some in the class pointed to computer programming and the need to work in different number systems, like binary and hexadecimal. True enough that understanding base 10, base 2, and base 16 can have vocational value, but what if you don't want to be a computer programmer? How, I probed, would you think about setting the timing for traffic lights to make sure the cars don't get backed up? Would you turn to literature? History? Science? It's about understanding and articulating a system with consistent rules, like our number system. Over time we're going to forget a lot of the specific math content we learn in school just as we'll forget the details of Tuck Everlasting and Hamlet because we just don't use them. What are the more general learnings we want students to carry with them from digging into the math?
The answers, in part, have been articulated for quite some time in the NCTM process standards and the National Research Council's Adding It Up report. The Common Core State Standards document has re-articulated them quite well in its Standards for Mathematical Practice. We teach math as a vehicle for helping students to:
- Make sense of problems and persevere in solving them;
- Reason abstractly and quantitatively;
- Construct viable arguments and critique the reasoning of others;
- Model with mathematics;
- Use appropriate tools strategically;
- Attend to precision;
- Look for and make use of structure;
- Look for and express regularity in repeated reasoning.
These practices have broad applicability. Learning math allows us to practice and hone them. I'm glad to see them in the common standards adopted by so many states. Sadly, we know from experience that having the standards doesn't ensure they become embedded in instruction and, more importantly, assessment. We won't value these practices unless we have some ways of explicitly tracking their development.
We have much work to do. I'm hopeful though. Nick's students in South Africa and my students at Harvard were both pretty psyched to finally get under the hood of the math they were doing. It's a nice and powerful 'aha' when you realize, "Oh that's how it works. Cool!"
(For some accessible background on counting systems, check out chapter 2 of the wonderful book Here's Looking at Euclid by Alex Bellos. You can get a preview of the chapter on Google Books.)
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