I’ve recently attended several meetings about the Common Core State Standards that have been adopted in NJ (along with 36 other states). The thought of the work required to implement the new standards is overwhelming, but I’m excited about the coherence and clarity that the standards will bring to math education. I realize there are plenty of people who disagree with the movement towards national standards, but I feel that this movement is best for the nation’s students. Shouldn’t all students be aiming towards the same goal? Shouldn’t all teachers have similar expectations for their students at each grade level? The standards aren’t taking away teachers’ autonomy. Instead, they are bringing more equity to math education and making our system more comparable to high-performing countries with educational systems that have proven to surpass that of the United States.
While I’m eager for the final outcome of these standards, I realize that the next few years are going to be a bit dicey. This week, NJ released a gradual release implementation plan, and while a 3-year roll-out is logical, there will still be challenges in terms of ensuring that students don’t skip content during transitional years. Expectations will require teachers to confidently deliver the grade-level standards while recognizing the need to include foundational content that students may have missed. While educators should teach to the standards, we must recognize that these standards are the core, but not the whole, of what must be taught. Teachers need to use their expertise and collaborate to ensure that students are provided with a comprehensive mathematics education. In several years, students will be fully integrated into Common Core Standard instruction. That’s the moment I’m waiting for – the time when students move through a coherent curriculum with limited repetition enabling a depth of knowledge that will allow for the appreciation of the beauty of math. A bit idealistic? Perhaps, but I’m one who likes to embark on new challenges with excitement about the outcome. Only time will tell if my eagerness is warranted.
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I received a press release from the journal Science that new research shows, "Quizzes don't just tell us how well we've memorized something—they actually help us remember it." It reminded me of a conversation while I was at Scholastic about how students often are so sure that they know how to do the math because they look at a problem, then the answer, and think, "yep, that makes sense." They never actually try to solve the problem without looking at the answer and then draw a blank on the exam. There's just something about having to retrieve things from memory that seems to help us learn, but what is that something?
In the study, English speaking participants were given Swahili words to learn. Some of the participants were given a quiz on the information before a final test a week later. Others were given extra study time in lieu of the quiz. It turns out that the group that was given the quiz performed three(!) times better than the study-only group. How this happened, the release states, "seems to be that we give ourselves more effective mental hints when we're being tested than when we're just studying."
Researchers Mary Pyc and Katherine Rawson "call
these mental hints 'mediators' and define them as words, phrases or concepts that link a cue to the 'target' that we're trying to remember. They hypothesized that mediators used during testing are more likely to be remembered and used effectively than mediators used when simply studying. During the initial study period, the students were asked to come up with mediators that looked or sounded similar to the foreign language cue and were semantically related to the English target. In the 'wingu-cloud' example, 'wing' might be the mediator." Students who were quizzed were better able to remember their mediators during test day.
I can remember several mnemonic strategies from math class, such as "Please Excuse My Dear Aunt Sally" for order of operations. Perhaps all those exams have drilled that into my brain to this day. It's important to note, though, that using mediators is just one explanation for why tests may help us remember things. Though there's strong research support that practice tests help people remember things, the explanations for why that is are being actively studied. For example, Science also reports in their online news that other research has suggested that testing enhances learning by helping students allocate study time to the most difficult-to-master concepts.
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We are happy to have Math Solutions – the newest member of the Scholastic family – contribute to the Math Hub blog.
There are many places that teachers can go for some good quality resources to improve their math instruction. Here are two of our favorites:
Great resources including searchable activities and lessons; interactive, online examples of NCTM's Standards; and a wonderfully useful collection of web links.
Free Classroom Lessons from Math Solutions:
Find new ways to approach a problem or a strategy that will help your students 'see' the answer. Also, submit your favorite classroom lesson to share with the community.
What other free professional development resources do you like? Please share!
Each week, every 3rd through 8th grade student in my district completes a state test-prep assignment that includes multiple choice and open-ended problem solving. I developed this program to help integrate test preparation throughout the year and also to track student progress. We’re six weeks into the program now which means that I’ve graded approximately 2,700 of these papers, and I’m starting to notice some trends.
First, in terms of the multiple-choice questions, student strengths are typically with the Number and Operations strand and the greatest weaknesses tend to be with either Geometry and Measurement items, or Probability and Data Analysis items. Clearly this is indicative of teachers’ likelihood to spend the most time on Number and Operations and less time on the other strands. While I do feel strongly that Number and Operations should be a focus because it is the foundation for all other mathematics, I continue to encourage my colleagues to find ways to integrate the other strands into their daily routines.
Yes, I’ve told them this before, but seeing student results on a weekly basis highlights the consistent weaknesses in certain strands. It makes students’ areas of struggle more explicit.
The most interesting results, however, draw from the open-ended question responses. Students tend to have a difficult time solving multi-step problems and explaining their answers. Many know how to find the answers – some can even solve parts of the problems in their heads. However, they lack the ability to fully explain their solutions. I’m a lover of math, and I haven't had the same passion for language arts. However, we must recognize that writing must be taught as a component of every subject. We must train students to provide a rationale for everything they do in math. Sometimes, I tell my students to pretend that they’re explaining a topic to an alien from Outer Space. This tends to help them realize that every step must be clearly explained. What do you do to help your kids realize that doing mental math without showing work isn’t always rewarded?
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Next week, I am hosting my district's 2nd annual Math-o-Lantern Night. I began this event last year as a way to get all 3rd and 4th grade students and their parents involved in a family math activity. The goal of the event is to help students and parents understand that, yes, math can be fun, and math instruction doesn't always have to be contained within the walls of the classroom. Students who sign up for this event are invited to the school one evening to participate in several activities that incorporate math and the art of pumpkin carving. Our Parent-Teacher Association provides the pumpkins and the parents are asked to bring a carving implement – permanent markers are provided for those who would rather not carve.
There are many mathematically-related activities to the event. First, students are asked to estimate the pumpkin's weight, height, circumference, and also the number of seeds inside the pumpkin. Then, students use various tools provided to find the actual measurements and reflect on the differences in their estimates and the actual numbers. By the way...don't underestimate the number of seeds in a small pumpkin. I was proud of my students predicting that their pumpkin would contain 60 or 80 seeds – until they opened the pumpkin up and started counting in the four- and five-hundreds!
With the help of a teacher-volunteer, students also created an x-y graph to compare the weight of the pumpkin with the number of seeds inside. Although this skill is a bit advanced for them, they could easily see the correlation from the visual display. After identifying shapes they would use to create their pumpkin face and sketching their design on a pumpkin, parents and students worked together to carefully carve their pumpkin.
Students left the event with their very own masterpiece, a new understanding of some math concepts, and the realization that math can be fun and rewarding. What more could I ask for in one evening?
You can download the full Math-O-Lantern worksheet here, and have fun using it in your classroom, too!
Last week, I provided information about Response to Intervention (RTI)
and details about how Tier 1 is meant to be implemented
to all students. While some remediation is acceptable in Tier 1, students who do not appear to be making progress should be provided with Tier 2 interventions, and one or two students may eventually be recommended for Tier 3 interventions.
Tier 2 instruction is typically provided for 30-40 minutes per day, several days per week. Interventions are often provided with the assistance of an additional teacher who comes into the classroom and works with the small group of students, or who assists the group of students in another setting. Tier 3 instruction is also provided in addition to the regular classroom instruction. However, Tier 3 instruction is often provided one-on-one because the number of students needing this level of intervention is quite limited.
The Institute of Educational Sciences (IES) has provided a guide, Assisting Students Struggling with Mathematics: Response to Intervention for Elementary and Middle Schools. This guide provides research-based recommendations regarding RTI, including tips for implementing Tier 2 and Tier 3 interventions. While several suggestions are made, below are the ones that have been found to have at least a moderate level of effectiveness.
- Instruction should be explicit and systematic – provide models, guided practice, and feedback.
- Interventions should incorporate instruction in solving word problems that derive from common underlying structures.
- Interventions should provide opportunities for visual representations.
- About 10 minutes of each intervention session should focus on the retrieval of basic facts.
While integrating an RTI framework is not a simple task for any school, I hope that you’ve learned a little more about what is necessary for the process to work. Please, let us know…how is your school working with RTI? Have your efforts been successful in improving student achievement?
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I spent part of this past weekend at a clicker conference at Harvard sponsored by Turning Technologies. Turning Technologies is a leader in the field of audience response systems, most likely familiar in the K-12 world in the form of "clickers", small handheld devices for capturing individual responses. Turning Technologies, though, along with other companies in the field, have extended the response technology to any internet-enabled device. I was feeling pretty geeky at the conference with a clicker, my laptop, my iPod Touch, and my Blackberry spread out in front of me, each one responding to the prompt on the screen at the front of the room.
I'm a regular user of clickers in the course I teach at the Harvard Graduate School of Education. My son, who is a sophomore at Harvard, had to buy a clicker last year for his physics and math courses, and I've seen a number of uses in K-12. I was struck, however, at the range of response system users I met at the conference from across disciplines and industries: an anesthesiologist from Yale School of Medicine; a trainer of medical devices; K-12 teachers; an undergraduate English professor; distance learning educators; a woman who supports conferences and training for the petroleum industry (conferences with over 80,000 participants!). Audience response is finding a home in a lot of places.
In general, I saw three main categories of use:
- Assessing learners. Picture a sequence of multiple choice questions projected on a big screen. Each student clicks in his or her answer with the system capturing and reporting results. This mass-assessment application of the technology gets used in K-12 and higher education.
- Assessing instruction. While assessing students can lead to adjustments in instruction, several of the folks I met were really interested in finding out if, for example, the presentation on new techniques for evaluating the potential of energy reserves in different geological formations was effective. Or do the sales folks now feel prepared to sell new medical imaging devices to hospitals? The assumption here is that the audience is motivated to learn. The participants need the knowledge to do their work. The question is: did the instruction provide it adequately? I do a bit of this in my course with an activity I call "Got It/Don't Got It". Students respond to peer presentations (and mine) with a simple up or down vote on whether they understood what was said and shown. The results can prompt a helpful re-articulation.
- Facilitating instruction. Several conference participants described how they use the clickers to help keep students engaged and to enlist anonymous opinions to prompt discussion. My main use of clickers falls into this category, mainly using surveys to activate prior knowledge and build a mental model for the topic we're covering that day. My son's physics teachers use the technology to foster a type of "think-pair-share". A well-crafted multiple choice problem elicits responses that are pretty evenly distributed across the choices (good question development is essential). The software displays that distribution, and that gets the students scratching their heads (think). They talk to their neighbors about their thinking and revise their choices (pair). New voting and sharing of strategies (share) illuminates errors, corrections, and different approaches to the problem. When it works, it works well.
If you're a response system user, feel free to share what you do here. If you're a critic, share those thoughts as well. Here's another technology with lots of potential. It's up to us to use it well.
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Now that I've settled into the school year and reviewed last year's test results, it's time for me to really dig into my role as a "consultant" to the teachers and students. As always, our goal is to ensure that all students are met at their current level and moved forward on their educational path. Thankfully, guidelines have been put in place to support this process. In my last entry, I discussed the basics of the Response to Intervention (RTI) model, and now I'm going to help you implement it.
Tier 1 of the model suggests that all students be supported through consistent monitoring and subsequent curriculum adjustments. This involves screening all students at the beginning of the year and monitoring progress throughout the year. The initial screening can give you a good idea about the "buckets" your students are likely to fall into. However, it's the consistent monitoring that is essential to track the progress of each student. As students are monitored, you might see some systematic issues with the majority of students. Oftentimes, these issues are with the curriculum and how this content is being taught. At this point, you should take a look at your curriculum and adjust teaching strategies to better meet the needs of all students.
You might also see some students doing very well on the assessment and others who appear to be struggling just a bit. This could indicate the need for more differentiated instruction within the classroom setting. It is important that students are given the opportunity to remain in the standard classroom setting, and many times this can be achieved by providing more challenging assignments to some and less intense assignments and support tools to others. Once these strategies are in place, give it a few weeks and reassess all students to determine if improvements have been made. If a small group of students are still struggling, they are likely to be candidates for Tier 2 interventions. Stay tuned for tips on how to help those students succeed in your math class!
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The President's Council of Advisors on Science and Technology recently released a 130-page online report suggesting changes to the education system to better prepare and inspire K-12 students to pursue STEM fields. One thing that I appreciated about this report is that the group (which included Google CEO Eric Schmidt!) suggested actionable items rather than doing a review of the literature and organizing general priorities, which seemed to be a common theme in the last half dozen or so reports of this nature that I have read.
A recommendation that particularly caught my eye is spending $325 million a year to boost the salaries of some 22,000 top-notch math and science teachers around the country. That comes out to an average of $14,772 for each, plus discretionary funds for the classroom, which is a big deal. However, what seems like an even bigger deal is that this means teachers would be assessed to be selected as part of this winning batch. I was scrolling down the report, looking forward to reading the recommendations of how to select these teachers, but unfortunately, this was it: "We recommend that the Federal Government undertake a rapid six-month study to address the issues in implementing a STEM Master Teachers Corps - including the selection process and criteria for the teachers and the organization and administrative structure."
The council would like the teachers to be "selected based on their demonstrated ability to prepare and inspire students." Now here's what I wonder: How can a teacher's ability to inspire students be measured? Thoughts?
photo credit: http://www.whitehouse.gov/administration/eop/ostp/pcast/docsreports
Have you ever asked your students, "What do you want to do for homework?" An article by Kathleen Cushman called Show Us What Homework's For gives several students' perspectives on how to make homework meaningful, and students make suggestions about how to make it less of a chore and more motivating.
The students quoted in this article are in high school, but their ideas can be applied to students of any age. Through vignettes from their own life, students share five suggestions about how to make homework valuable. You'll notice that some of these overlap with suggestions provided by the "experts" in terms of what makes a good homework assignment.
- Purpose – Homework should match students' individual weaknesses. Just as athletes spend time refining the skills they struggle with, students should be given opportunities to work on their stumbling blocks.
- Follow Up – Don't just take a cursory look at assignments, really try to understand where students went wrong and provide opportunities for students to revise their work.
- No Grading – Homework is meant to enable students to practice the skills they have not mastered. If it's graded, students will find the correct answer using any means possible without actually grasping the concept.
- Better Use of Time – Assign less homework but gear it towards deeper understanding. Doing the same problems over and over again just creates routine, not comprehension.
- Follow the Four R's – Readiness, Repetition, Review and Revision.
Finally, students made suggestions about specific activities that can help them master content using the guidelines above. Give one of these ideas a try...you might be pleasantly surprised with the outcome!
|In This Learning Situation...
||Instead of This
|You introduced new material in class.
||Assigning a question set so we will remember the material.
||Ask us to think up a homework task that follows up on this material and to explain our choices.
|You want us to read an article before a class discussion.
||Making us answer questions that prove we read it.
||Ask us to write down two or three questions we have after reading the article.
|You want to see whether we understand a key concept (such as literary irony).
||Making us complete a worksheet.
||Ask us to demonstrate the concept for the class in small groups, using any medium.
|You want us to see how a math procedure applies in various situations.
||Assigning 10 word problems that involve this procedure.
||Ask small groups to choose one word problem that applies this procedure in a real-world situation, solve it, and present it to the class.
|You want us to memorize facts (such as dates in history).
||Handing out a list that we will be tested on.
||Ask each student to share with the class a memorization trick (such as a visual cue) that works with one item on this list.
|You want us to remember what you taught last month.
||Assigning a review sheet.
||Give frequent short pop quizzes about earlier material. Go over each quiz, but don't count the grade.
Source: Show Us What Homework's For by Kathleen Cushman
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Most teachers have an understanding of the term Response to Intervention (RTI) when it comes to Language Arts. Over the past several years, however, there has been an increased emphasis on RTI in the mathematics classroom. In upcoming blog entries, I'll be offering tips on how to successfully implement RTI in your classroom, so I thought I'd start first by refreshing everyone's memory about the purpose of RTI.
The goal of RTI is to promote continuous monitoring and intervention to prevent students from falling behind. Implementing RTI effectively can help differentiate between students who have specific learning disabilities and those who are struggling in a particular area but can be brought up to level with some intervention. The three tiers of the RTI model are meant to be flexible so that students can receive the appropriate level of support based on their needs at a particular time. Students who do not respond after moving through the RTI framework may receive additional services for a specific learning disability.
TIER 1 is the most comprehensive tier that covers standard classroom instruction and review for all students. In Tier 1, students are screened, monitored and should receive research-based instruction. This instruction may include some remediation, but students receiving only Tier 1 instruction are considered to be working on grade level.
TIER 2 interventions are designed for students who have been identified as struggling with a particular content area.
TIER 3 interventions include specific, intensive support for students who may be working several years below grade level. Tier 3 students often receive one-on-one help and are closely monitored to determine if improvements are being made. If interventions offered in Tier 3 are unsuccessful, students should be afforded the opportunity to receive additional services through the school's special education department.
Now that we've reviewed the Response to Intervention framework, look for future posts about how to deliver math instruction at each level. In the meantime, share your school's sucess stories with implementing the RTI framework for math instruction.
Just this past month, the U.S. Department of Education published a practice guide called Developing Effective Fractions Instruction for Kindergarten Through 8th Grade. It is definitely worth the read if you or your teachers are experiencing some challenges in teaching fractions.
The panel arrived at five recommendations for educators to improve students' understanding of fractions:
Recommendation 1*. Build on students' informal understanding of sharing and proportionality to develop initial fraction concepts.
Recommendation 2*. Help students recognize that fractions are numbers and that they expand the number system beyond whole numbers. Use number lines as a central representational tool in teaching this and other fraction concepts from the early grades onward.
Recommendation 3*. Help students understand why procedures for computations with fractions make sense.
Recommendation 4. Develop students' conceptual understanding of strategies for solving ratio, rate, and proportion problems before exposing them to cross-multiplication as a procedure to use to solve problems.
Recommendation 5*. Professional development programs should place a high priority on improving teachers' understanding of fractions and of how to teach them.
(*) These recommendations are supported by our newest program, Fraction Nation.
Download the full PDF of the practice guide
Do you think there is anything missing from this list?
What recommendations do you have for effective fractions instruction?
photo credit: http://ies.ed.gov/ncee/wwc/
Several weeks ago, I mentioned that one of my major goals for the school year is to work on our district's homework policy. Too many students' grades are suffering because they are not completing homework, and I can't help but think that part of this has to do with the quantity and quality of the homework assignments. Recently, I came across an article titled Five Hallmarks of Good Homework, which details the characteristics of a meaningful homework assignment:
- The assignment has purpose. Too often, students are given assignment that require students to do a lot, but the work doesn't have a meaningful goal. Purposeful tasks are meant to help students master a skill, not struggle with a concept they did not fully understand during class. The author suggests distributing small doses of homework over time. For example, a student may complete "three questions or problems to check for understanding of today's lesson and 10 questions or problems to practice previous learning."
- The assignment is efficient. Rather than giving students "busy work", students should create projects that truly show what students know. Creating a review game or educational video reviewing a topic can be engaging and can also measure student understanding.
- The assignment gives students ownership. Students consistently complete assignments that are teacher-driven. Instead, give students more ownership of their work. Ask students to study their multiplication table in the way that works best for them, and ask that they document their method and reflect on its effectiveness.
- The assignment makes students feel competent. "Homework that students can't do without help is not good homework." Be sure that assignments are appropriate for students' ability levels. Provide struggling students with tools to help them complete assignments. An average student might be successful with open-ended questions, but perhaps a struggling student needs the support of multiple choice or just fewer problems. All students should complete homework in roughly the same amount of time, so try to give each student the tools to make that possible.
- The homework should look interesting. Students are going to be more motivated to complete homework that is visually uncluttered and looks interesting and inviting. Try to add interest to assignments by giving students enough room to work, and adding graphics where appropriate.
By considering these five hallmarks when creating and assigning homework, you can be sure your students' time is spent in a meaningful manner to promote growth and understanding.
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The film Waiting for Superman, which follows five families in desperate searches for quality schools, has quickly generated interest, passion, and controversy. I attended a packed preview screening last week hosted by the Harvard Graduate School of Education. The buzz in the informal conversations that naturally occurred during the ensuing reception captured the gut-wrenching frustration at watching the futures of lovely children and caring families rest on the roll of a lottery ball or the random selection of a name out of a box or computer. Why can't these kids get the kind of quality education they desire? It's just not fair.
At the same time, many in the audience, myself included, felt a bit cheated by the overly simplistic message about what needs to be done. "The path is simple," reads the text as the credits roll. Really? Just knock down the unions and bureaucracies, open the door to charter schools, and great teachers will flow into the classrooms? While the stories of Geoffrey Canada's Harlem Children's Zone and the growing KIPP School phenomena are inspiring, the film admits that only 1 in 5 charter schools seem to do better than the public alternative. Getting rid of bad teachers is a great idea, but where will all the great replacements come from? It's certainly heartening to see growth in popularity of Teach for America, the number one employer on several top college campuses last spring, but it alone can't meet the need. Will merit pay, an approach that gets implicit endorsement in the film, provide the incentive? As someone who believes in following the research, I'm skeptical. A recent report from the National Center on Performance Incentives at Vanderbilt University found that merit pay alone had no impact on student test scores. The report recommended more nuanced solutions.
Okay, I'm willing to forgive the oversimplifications in the film if it indeed generates attention, constructive discussion, and, most importantly, thoughtful, individual action. The big message that we shouldn't wait for Superman to fix the problem is right on the mark. We can and should be doing something. I've talked to folks who, after watching the film, considered for the first time becoming teachers. Others expressed new energy ready to be tapped. A cultural groundswell that attracts large numbers of the best and the brightest into teaching, a characteristic of the highest performing nations, would be fabulous (I recommend reading the McKinsey report on the world's best performing schools).
And let's not forget that good things are happening, in traditional public schools and charters. The Achievement Gap Initiative at Harvard, for example, captured wonderful success stories of 15 public high schools from across the country. It's not everywhere, but each success is further proof that it can be. And to be honest, it's gratifying to find that our programs – READ 180, System 44, FASTT Math, among others – are often part of those turnaround stories. We're doing what we can now. We're not waiting for Superman. I hope you don't either.
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There are lots of debates in education — some include more subtle teaching ideas, others get teachers fired up and inspire passionate discussions. One recurring debate among educators that spans grade levels and includes both traditional and progressive teachers involves the appropriate use of calculators in math class. Just the word calculator is likely to bring an immediate opinion to your mind.
I'd like to share with you my thoughts about calculator use for regular education elementary students. I feel strongly that just as some students need glasses or a hearing aid, there will always exist students who need a calculator due to a diagnosed learning disability. However, most students should not have free access to calculators. Students should be introduced to a calculator as one tool in their tool box, just as their brain or a pencil and paper are tools. Hence, they should learn when the tool is appropriate to use and that, at the elementary level, there is no topic that requires calculator use. Students need to master computational skills by hand and using mental math before they are introduced to the calculator. Additionally, students need to get into the routine of estimating answers prior to performing any type of calculation. One of the biggest issues with calculators is that human-error is often not considered. Students feel that if the number shows up on the calculator, it must be correct. They don't consider that they may have typed a number in the tool incorrectly. Promoting estimation will help students recognize if an answer makes sense.
There are some activities that are calculator-appropriate when effectively introduced in the elementary classroom. Go ahead and introduce a pattern investigation using repeated operations to help students build patterns greater than they are capable of by hand. Or, have your students check their own work using a calculator — just be sure they use the tool after they've completed the computation by hand. However you choose to use a calculator, be creative. Just remember, students should not rely on the calculator for any computation that they can complete using paper and pencil!
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