Good afternoon and welcome to this month's chat with NCTM President Cathy Seeley. Our first question is from
This is a really complex issue. When is teaching to the test appropriate?
Does the NCTM have a list of state mathematics standards and assessments that they “approve” of? In other words, which state standards and assessments are doing a good job of highlighting “important mathematics”?
Teaching to the test might be appropriate under some situations, although more appropriately, I might say that testing can be part of a cohesive mathematics program that supports student learning. The test would need to be a good measure of the mathematics identified by the state or district as important (this is still a local/state decision). However, one danger is that even a ‘good’ test on a ‘good’ set of standards can be over-emphasized. There are many examples of wonderful open-ended problem solving test items that have been practiced to the level of becoming routine. Teaching mathematics occurs within a complex system that includes state standards, accountability testing, curriculum, teaching, classroom assessment, and other factors. All of these factors need to be considered in determining how to best improve student learning.
NCTM does not endorse programs as our practice. Some work on looking at state standards is currently being done by the Center for the Study of Mathematics Curriculum. Also, the Council of Chief State School Officers collects data on state practices in math and science, but I don’t specifically know of work done in the areas you mentioned.
I just wanted to make a quick comment, not a question. I will be student teaching in a few weeks and I was just saddened to find out that the school I will be teaching at has PSSA Fridays ( Pennsylvania state tests). On Friday, teachers are required to focus solely on state tests. How sad it truly is that there needs to be a day set aside every week.
What a cost it is to set aside 20 percent of our instructional time for this purpose, rather than allowing/supporting teachers to teach a good mathematics program and also prepare students for the test.
Greensboro, North Carolina
An aspect of this issue worth considering is building and district level leadership: What can our profession do to educate the implicated leadership on the merits of a good mathematics program? Teachers complain that they are being squeezed towards “It is all about the test.”
Building and district-level leadership is definitely important in setting direction. Unfortunately, administrators experience pressure from many sources, including the community and higher-level administrators or policy makers. Educating administrators about quality mathematics teaching is worth the effort. This is an appropriate avenue for collaboration and outreach. Perhaps mathematics teachers, or representatives of local mathematics teacher professional associations, can volunteer to make presentations on mathematics at administrators’ conferences or district principals’ meetings.
I would argue that “teaching to the test,” as it implies in its everyday sense, is an indicator of inadequate professional development for classroom teachers, especially math teachers. While I am against “teaching to the test,” I was amazed the other day when I observed a fifth grader teacher who actually took a data analysis problem from the test prep book, and converted it to a very meaningful activity. Based on the setup of the original problem, she collected some favorite-books data from her class, and paired up the kids for graphing. Seeing the kids so engaged in their colorful poster work, and most important, how the teacher asked each group to present their graph to the class and answer questions, I couldn’t help thinking that the teacher plays an essential role in “teaching to the test.” Given the test is here to stay for quite some time if not forever, I would think teachers, especially those lacking in experience with the right way to “teach to the test,” need to understand the test, adapt test problems or relevant topics to her kids. In other words, recognizing the existence of the test, teachers need to be shown how they could teach to the test in the best interest of mathematics learning and the long-term growth of their students. Most of the test problems afford great opportunities for rich classroom discussions and cover important strands of standards, but the teacher has to see them in such way as to teach to the mathematics rather than to the test. What do you think of it?
This is a wonderful example of putting testing and test preparation into perspective as part of a complete, rich balanced mathematics program. What a wonderful example of helping students not only learn mathematics, but potentially see beyond a potentially limited test item to the mathematics behind it. Thanks for sharing this appropriate way of teaching for test achievement without sacrificing mathematical learning.
The issue of testing what’s important versus testing every objective in the curriculum has led to test fatigue and other problems. Has NCTM responded to the 2 days of testing for example we “suffer through” in Georgia?
NCTM has not specifically addressed the issue of multiple-day testing. I would recommend, however, the new position statement from NCTM on High-Stakes Tests. In general, I would say that testing has to be put into perspective as part of the total mathematics program, always considering cost-benefit issues. How much instructional time is lost in order to get what quality of results? Are students demonstrating the best they can do for any given situation? What level of information does the test provide both teachers and students? And many other related questions. Large-scale tests are only snapshots, so it is especially important to not have them overpower the heart of the mathematics program—teaching and learning in the classroom.
Bronx, New York
Do you see a national test anywhere in our future?
What an interesting question! In a past life, I was part of a group asked to work on a voluntary national math test for eighth-graders. In my opinion, this was a good idea and headed in a nice direction. Unfortunately, it got sidetracked by a change in administration. But I again hear rumblings about something along this line. Such a test, if it were well designed and not too intrusive (as in not multiple days for any student) might be a move toward identifying national priorities. Even so, it would have to fit in with the broader picture of the mathematics program. In particular, I would hope that any state or district choosing to participate in such a test would have it replace, not add to, their current testing for whatever students participated. And, as with my other comments, it really needs to be weighed both nationally and locally in terms of potential costs and benefits. Testing is important, but it cannot drive our teaching in inappropriate directions.
I question why “clue words” or I call them key words, should not be emphasized. When students have a word problem should they be picking out the key words to know how to set the problem in an equation format to solve?
There are a couple of problems that I see with using key words or clue words. For most key words, many of us could make up problems that might use the given key words in different ways, not always calling for the same operation. For example, some elementary students look for the words “in all” or “all together” to mean that they should add up the numbers in the problem. But the words might be used differently in a problem (A truck just delivered enough desks to be placed in all the classrooms in the new school building. They unloaded 80 palettes of 6 desks each. There are 16 classrooms in the building. How many desks will be placed in each classroom?). This may seem contrived, but I have seen too many students who incorrectly remember what the words stand for.
There is nothing wrong with students knowing that the words “in all,” when used to describe a collection of things, often means to put groups of objects together. But this is based on the meaning of the words and the meaning of the operations, and it is very different from asking them to memorize tricks like “when you see the words ‘in all,’ you should add.’”
But of most importance, taking time to memorize tricks and lists of words might take away instructional time that could be used to learn the operations well. In my opinion, the goal should be for students to think about the mathematics involved and perform well on the test based on their understanding.
Several questions expressed similar points of view about teaching to the test. Below are several of these submissions:
Hamilton , New Jersey :
I have been saying this for years! If your curriculum is well balanced, and you are enriching the curriculum with special activities, there is no reason to teach to the test. Be sure the students know the basics, and they should do fine. You shouldn’t have to take time out of teaching the curriculum to teach to the test. There’s not enough time in the school year to begin with, let alone adding this.
Lithonia , Georgia :
I agree with you, “The best preparation for any test is teaching a good mathematics program well to every student.” I am fed up with school districts admonishing their teachers to teach to a test.
Rockford , Illinois :
Teaching to the test becomes drill and practice with very little critical thinking. I am pushed to cover the topics for the test and lose the time for global problems. They are the problems that students will remember. How can we change this short sightedness?
I also teach math in intensive block. It is toooo fast for most students! How can we get rid of it?
It sounds like many folks are dealing with similar frustrations. It takes courage and persistence to stick to the goal of teaching mathematics well as the best preparation for the test.
Grand Rapids, Michigan
In Michigan, we’re MEAP testing in October! I would like to know if any other state in the United States has its state tests at the beginning of the year. (The premise was that Michigan teachers waste too much time reviewing at the beginning of the year.) Districts were supposed to test in October and be given the results in December, so that Michigan teachers could see the results and plan for half a year. I would argue that Michigan teachers would rather plan for an entire year for their students. Here it is February and the test results have not been released to districts. At a conference last year, I asked the state official who changed the testing time to October why the Michigan MEAP test wasn’t given in April or May, after teachers have taught all the benchmarks or strands. When I commented that maybe university students should be tested 4 or 5 months after they’ve taken their courses, the state official that moved the testing to October, stated “Oh! The university won’t do that!” Now my question is “Why do it to elementary students?” So could I get a list of any other states that test at the beginning of the year, or a list of all states’ testing dates? Thank you!
This is always a challenge. Several years ago in my state of Texas, the state test was given at the beginning of the year for a few years (it is now given in the spring again). I kind of liked this practice of early fall testing, since it tended to disrupt instruction less than having it later in the year. Part of the problem is that tests are rarely given at the true end of the year. More often, they are given two-thirds of the way through the year, or at least several weeks before the end of the school year. Yet, these assessments are often viewed as measuring the year’s learning. In too many cases, this means that teachers rush to cover the content that will be tested before the magic date, and then that they have several weeks left over that seem far less important. There is no good time of the year to administer a high-stakes test, so it comes down to cost and benefit again, taking into account how the test can best support mathematics teaching and learning.
This would be useful data to gather, and perhaps the Council of Chief State School Officers (the organization of state-level commissioners and superintendents) or one of the National Centers for Learning and Teaching might choose to gather this data if it is not already available.
NCTM advocates teaching Algebra 1 in grade 8. The State of Missouri has developed Grade Level Expectations (GLE). In mathematics, students are held accountable for these on the state assessment in grades 8 and 10. The dilemma occurs when 8th-grade teachers have to decide whether to cover the GLE or the Algebra curriculum, because they don’t have time to do both. If they leave out the GLE, students will not fare well on the test. If they leave out Algebra 1 curriculum, students are ill-prepared for the upper level math they have ahead of them. Any suggestions?
I am surprised to see you say that NCTM advocates Algebra I in grade 8. Throughout last year, the Council focused on how to develop algebraic thinking from preschool through high school. You might also look at my President’s Message from March 2005 called “Pushing Algebra Down” and the transcript from the online chat based on that President’s Message.
In any case, your question probably needs to be answered locally in light of your state and school system’s decisions about what content should be addressed at each grade level. If there is a mismatch between middle school expectations and high school expectations, this should probably be addressed at the state and district level, since a major consideration in developing state standards and the district curriculum is how they develop and connect across the grades.
Our school has identified math as an area of focus this school year. We are currently working with a professional math consultant on developing understanding of the mathematics we teach and teaching for student understanding. We have struggled for years with the same problem: students not retaining important mathematical concepts and teachers always playing catch-up with students who come to a new grade level without the foundation needed to be successful at that level of development. We have addressed many of our causes and are now working toward a solution. Unfortunately many of the causes are ones that are out of our control (i.e. high transient rate, low socioeconomic background, lack of parental involvement, lack of motivation, and emotional/social needs). Many of our teachers are on board and trying to put forth the extra effort to change and implement a well-balanced mathematics program. But here’s the problem—now that testing is at hand and we have spent so much time trying to back up, catch up and “uncover” student understanding we feel that we have not prepared them for what will be presented on the test as being age appropriate for them. All growth cannot be made up in one year. We realize this is a long-term commitment. We’ve learned from this process that understanding (of any concept) exists on a continuum. We know that all students are not going to be at the same place at the same time. Therefore, do we ditch everything we’ve tried to do as far as slowing down and teaching for understanding, or rush through the curriculum just to get it covered for the test? We know the answer is “No” if we want to see continued growth over time, but it gets increasingly more difficult as the testing window draws nearer. We are interested in hearing from other educators who are faced with this dilemma, as well as answers to the questions you posed: “How can we balance teaching good mathematics and preparing for the state test?” and “Are there effective test-preparation strategies that support student learning?”
We would also like to consider these questions:
What can we do to have our teaching drive the test rather than the test drive our teaching?
What are colleges doing to prepare new teachers to teach the student not the test (i.e. a lot of experienced teachers have been taught the “multiple choice” way.
You are getting at the heart of the issue about the dilemma many teachers face—whether to teach the mathematics they know their students need or prepare for a test on what might be inappropriate mathematics for where the students are. You might guess that I come down on the side of teaching the mathematics students need, with a very watchful eye on teaching more than a year’s worth of mathematics in a year for students who are behind. This means additional time and support for students through double-blocked periods, after-school programs or summer programs. One way we got to the situation of having so many below-level students is by allowing students who are behind to move ever more slowly through the curriculum.
That said, it sounds like you are on the right track to bring these students up to a reasonable level of achievement. I continue to believe in teaching the mathematics, rather than the test. But perhaps there is a small compromise one could make. Perhaps it is possible to build into beginning of class warm-up time carefully selected items similar to some that that might appear on the test, provided the mathematics involved does not confuse students. In many cases, if we are developing conceptual understanding along with proficiency and problem solving in our primary mathematics program, students can transfer what they are learning to new situations that can even be approached as a puzzle or a challenge without spending significant time on them. Developing students’ ability to represent mathematics in different ways (graphs, tables, pictures, symbols) and developing algebraic thinking and reasoning are some examples of important mathematics that crosses grade levels and set students up for success in such challenges. But I hope that I would never take significant instructional time to try to teach inappropriate mathematics that might confuse students rather than advance their learning. I would rather prepare them to be somewhat frustrated on the test, than to be totally lost in their learning. Perhaps if we make the commitment to focus on the mathematics, the students will pleasantly surprise us on the test, even if we think the test is inappropriate.
What suggestions do you have for a teacher who has a number of students well below grade level in their class, but those students are still required to take a state exam? For example, an eighth grade student who does not know multiplication facts.
I would echo some of the comments from the previous question. But unfortunately, many eighth-graders do not know their multiplication facts. It is a challenge to decide how much instructional time to devote to these facts balanced against teaching some of the rich mathematics found in some middle school curricula and developing problem solving skills. I have written in previous chats about a former student of mine who seemed unable to learn fraction operations until she experienced success in algebra. She then learned her fractions. So for some students, it may come down to a teacher making a decision about what level of previous knowledge is essential in order to move on. This is important not only for testing, but for actually progressing in mathematics learning. Practicing multiplication facts is something that can also be done outside of math class time, possibly involving families.
Paterson, New Jersey
It’s been my experience that the teachers who are pressured to teach to the test are those in lower performing schools in urban settings. My colleagues in better performing schools don't emphasize the test and their students consider them just something they do on a day in March. Is this observation generally true?
This is an interesting hypothesis. I don’t know of anyone who has looked at this, and I honestly don’t know how true it might be. On one hand, I would expect that schools with traditionally high performance might experience less outside pressure to do things differently. However, I have spoken with many teachers from such schools who feel as much pressure as those in urban settings. Many of our urban schools face tremendous challenges that are not seen in suburban settings.
It could be a positive thing for any student to view the test as “something they do on a day in March,” provided they have just a little bit of healthy anxiety that causes them to prepare for the test. By this I mean common sense preparation such as getting a good night’s sleep, eating breakfast, being familiar with the format of the test, and knowing what to expect in terms of the mathematics that will be tested. But this again means focusing on the role of such tests as one part of a comprehensive math program, and not as the primary goal.
Our AP students have picked up on the strategy of the teachers being required to spend too much time on test preparation. How do we keep our advanced students from becoming the victims in this emphasis by the administration to teach only to the test?
Even our most advanced students can fall victim to this practice of teaching primarily around testable items. And even a quality test can be used inappropriately. The key will continue to be a personal commitment to teach the mathematics, and know that the test results will follow. Making this commitment is increasingly challenging in light of pressures to do otherwise.
Great message, as always. I will say that I think that we are moving in the right direction but do have more to do. Since many tests 15 to 20 years ago were much more based on only “solving” a calculation problem in no context whatsoever, we have at least allowed teachers and students to see that math is used in a context in real life and should be taught and tested primarily in that way. While this has moved us in the right direction as far a having students realize relevance to their real world it sometimes has allowed for the “tricks and clues” to become too much of the stressed part, more than the thinking and conceptual understanding and number sense. I believe that we as mathematics educators can continue the move toward these areas and have an influence in our schools to stress these ideas but also have an influence as the state tests are written to try to add some new items that really require thinking that does not rely on learning and implementing tricks. Many systems are now working on having students use “Exemplar” or algebraic-type problems at the elementary level that make students think and which actually cannot be worked by just using “tricks or clue words.” The more this is done and the more we move in the direction of using these type items as test items the more the teaching and emphasis in the classroom will move in that direction. Even though overcoming previous mindsets is difficult, it is not impossible. I’m a living testimony to that. Even though people tend to teach the way they were taught, I know that it is possible to help people see it differently, so we cannot become discouraged. It is so exciting to have teachers have “Ah ha!” experiences themselves about how they teach math. I have been there and done that personally and am now spreading the word as are many others. Things are changing and I believe if we keep letting our voices be heard and setting the example that the teaching and learning and perceptions of mathematics will continue to improve. Let’s continue sharing and setting that example for others.
Have a great day and thank you for all you do to set that example and share thoughts that make teachers and others THINK!
Classroom assessment is a critical component in a well-designed mathematics program, and improving our practice in this area can have multiple benefits. You have identified a couple of key points: that appropriate assessment can support student learning, and that educators can (and should) use what we learn to influence even state-level assessments. Thanks for sharing these professional thoughts.
As you know, the key to blending “teaching the test” and “teaching the curriculum” is to insist that the test be based on the whole curriculum, and not just a minimal subset of skills. There is no honor in passing what everyone knows is a basic skills test. Hold students and teachers alike accountable for the full curriculum. Success on such a basis is something students, the school, and the community can truly celebrate. Further, when the testing and teaching content match, no time spent on test preparation is wasted time in the classroom.
A colleague has said that we need to be careful about believing our own rhetoric. In terms of state accountability assessments, this is particularly dangerous. Many states have either phased in accountability tests that start off with lower level skills, but over a period of years increase the level of difficulty. Others have implemented fairly low scales to identify “mastery” or “proficiency,” with the intention of gradually raising the bar as students achieve higher scores. In neither case should we be complacent when many students reach these artificially low levels of achievement. Interpreting test scores should always involve a consideration of the level of the mathematics and the scale used to determine scores. Thanks for pointing to this other danger in dealing with large-scale assessments.
Tuxedo Park, New York
Do you think a separate class in test-taking skills is appropriate for a middle and/or high school? We are considering adding this course to alleviate the need the math teachers feel to teach toward the test?
That would entirely depend on what is included in such a course and what students would miss by taking it. I have a hard time understanding what kinds of test-taking skills might take a semester or a year to address, and I have a hard time thinking that this would motivate students to do well on the test. But if such a course were well-designed to focus on how to apply what you know and show all you know on a test, and if there were features related to study skills and/or reinforcement or application of knowledge learned in the content areas, then it might be worth considering. In general, my thinking would be to trust the professional teachers in the school to weigh the costs and benefits of such a move.
You have certainly hit a hot button here. What do you do when the administration is not only telling you to stop all other mathematics instruction to prepare for the test, but is also penalizing you (or so the threat goes) on your evaluation for doing otherwise?
Maybe this is a use for the NCTM News Bulletin. I know some people like to use these President’s Messages in support of their work (whenever they agree with what the President says, that is…). I also think that as mathematics educators, we need to find opportunities to work with administrators on what quality mathematics teaching looks like. You might also want to look at the NCTM position statement on Highly Qualified Teachers.
I often have my preservice teachers read your messages as a discussion starter because, try as I might, I couldn’t do a better job of supporting the same idea. Thank you for your willingness to speak up about such important, but not always popular, topics. Are all of your messages compiled and available at any one place? I have saved a few, but would like to go back and reread (and perhaps use) those that I didn’t save.
The President’s Messages can be found on the NCTM Web site at President's Messages.
We'll conclude by combining several comments that were submitted in advance.
I agree wholeheartedly with your essay on the misguided strategy of “teaching the tests.” I am currently a 37-year-old student teacher, teaching high school Geometry. In my classroom, I am required to have my students take “practice benchmark exams” and have been told numerous times that “most of these kids are never going to use this after graduation.” I would love to see your message taken to every school system and teacher of mathematics so that our children can receive the quality of mathematical education that enables them to develop the skills that I have found so useful throughout my life in the private sector. This is the reason I am leaving the private sector and entering the teaching profession, to use my experience with applied mathematics to entice children to see the value in and learn these skills. Thank you for so eloquently stating our position.
Rogue River, Oregon:
Multiple choice problems do not promote solving real-life mathematics problems. Four choices are not offered to the do-it-yourself home owner when needing to calculate the number of gallons of paint or yards of carpet needed to decorate a home. Mathematical reasoning and communication are necessary tools in solving real-life math problems. State assessment tests do not promote this type of learning.
Thank you! We’ve been saying this for such a long time and the powers that be NEVER listen!
Our system is trying to drive curriculum with tests—bad idea. There is tons of pressure. Our state test for 8th-grade students leads us to spend too much time, not getting students ready for Algebra in 9th grade. The results floor me—I have students who prove over and again each year in writing, speaking etc. that they know mathematics, yet score lower on the state multiple-choice test, than students who have shown me all year they cannot do the simplest mathematical problems accurately. How does one go about helping people understand that the test itself is not testing the material that every 8th-grade student should have mastered by 8th grade? And how do you get administrators to see these scores for what they really are?
New York City:
I think there is nothing wrong with test preparation as long as it is not overdone. Drilling students with practice test questions is wrong, but reviewing concepts and skills to foster a deeper understanding to prepare for the test is fine. Students have to become familiar with the types of things that are going to be asked of them.
These comments may prove useful to readers. For me, the bottom line is to put testing in its appropriate place—as a tool that can help teachers and support student learning. If any test cannot support appropriate mathematical learning, we should seriously consider its purpose. It comes down to costs and benefits, but what it really comes down to is good mathematics teaching and reasonable consideration of how to best help students show what they know on the test.
Thanks for this great participation in this chat. Let’s keep these discussions going at the school, district, state and national levels.
Thank you all for your participation today, and for those of you who submitted questions in advance. We had many more questions and submissions than we could address in this hour.
Please join us for the next online chat, on Cathy’s President’s Message in the March NCTM News Bulletin on technology, which will be at 4:00 p.m. EST on Tuesday, March 21.
“Teaching to the Test” (January/February 2006)
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