Which Varies in Your School: Instructional Time or Student Learning?
By Matt Larson, NCTM President
September 20, 2017
The most frequent questions I receive concern instructional time: How much time should we have for math class at the elementary level? Middle level? High school? What does the research say about time? Does NCTM have a position statement on time? The last question is the easiest to answer: NCTM does not have a current position statement on instructional time. Members with long memories might recall that once upon a time, NCTM did have a position statement on time, Math Takes Time.
I think these questions all falsely presume that there is an answer to the time challenge. In other words, the questions assume there is a magical fixed amount of instructional time that will maximize learning for each and every student. As much as we might want the research to tell us that exactly 61.2 minutes of daily math instruction leads to optimal student learning, it simply isn’t the case.
It is critical to note that how instructional time is used is as important as the fact that it is allocated. If students have additional instructional time but that time focuses on remediation, low-level skills, procedures without understanding and problem solving, or simply repeats strategies already explored in the classroom, then the addition of instructional time is unlikely to support student success. We need to grapple with how we organize and offer instructional time and what we do with it.
Ideally, instructional time would vary to meet the needs of students. For too long the amount of instructional time has been fixed and student learning has been unequal. If we are committed to ensuring equitable outcomes, then instructional time must be the variable factor. Students do not all learn at the same rate.
One of the most well established findings from research concerns opportunity to learn. If students do not have access to sufficient instructional time, their mathematics learning will suffer. And small amounts of time can make a big difference. For example, consider elementary instructional time. Suppose School District A allocates 50 minutes a day for daily math instruction in grades K–5, while School District B allocates 60 minutes a day. At first glance this may not seem like a meaningful difference, but the cumulative impact of this small daily difference is very significant.
Let’s assume both school districts have a 180-day school calendar. Over the course of a school year, students in School District A would need to go to math class an additional 36 days in order to receive as much math instruction as students in School District B. Over six years of elementary school, the students in District B will have received roughly a year more of math instruction than the students in School District A. Differences in instructional time clearly can contribute to inequitable student outcomes.
And the differences can be even more dramatic depending on how schools utilize instructional time to promote student learning. One of the most effective instructional supports is one in which student competence is carefully monitored on the basis of common formative assessments, and then the results of the assessments are used to target groups of students who receive additional instruction on the concepts and skills for which they have yet to reach proficiency.
The critical point is that the focused instruction takes place in addition to whole-class instruction rather than in place of it. Students are not removed from grade-level or subject-based instruction to receive targeted support. In too many cases, traditional interventions fail because they are not done to supplement whole-class instruction but instead of it.
Effective instruction ensures that each and every student has the opportunity to learn grade-level content (or above), while simultaneously guaranteeing each student the additional instructional time and support he or she needs to learn it. Highly effective schools are willing to allow time and support to vary in order to meet student needs.
At the elementary level, for example, all students might have 60 minutes dedicated to daily whole-class math instruction, and also have access to an additional 30-minute block of time designated for targeted mathematics instruction in areas of student need. If teachers work collaboratively in professional learning communities, they can fluidly regroup students among themselves during this additional time to more effectively re-engage students.
At the secondary level, schools that are committed to equitable outcomes often create “double-dose” courses, for example, two-period algebra courses or fluid support courses where students move in and out of a second period of math as the need arises. In both cases the support is directly linked to the grade-level or subject-based curriculum.
Where do you get this additional time? It can be done! Schools all across the country are doing it—but it requires strong leadership and a commitment to equity. Highly effective schools prioritize the curriculum and time, recognizing that while all subjects are important in school, some are more important than others because they are instrumental to student success in everything else. This makes English (including reading) and mathematics the most important subjects in school. After all, there is very little students can access in school if they cannot read. And a lack of quantitative skills will severely impede student success in science and many additional subjects as more and more issues are examined through a quantitative lens.
In Principles to Actions, NCTM argues that “Support for access and equity requires, but it is not limited to, high expectations, access to high-quality mathematics curriculum and instruction, adequate time for students to learn, appropriate emphasis on differentiated processes that broaden students’ productive engagement with mathematics, and human and material resources” (p. 60). If we are committed to ensuring that each and every student is successful in mathematics, then we must ensure that each and every student has adequate time and support to learn mathematics—including additional targeted and effective instructional time.
Varying instructional time instead of student learning by shifting the pedagogical focus from teaching to student understanding is one strategy we can leverage to create a culture of equity. I encourage you to engage with your colleagues and building/district leaders to examine not only how much instructional time you provide students but also to take action on the critical question: does instructional time or student learning vary in our school?
At my school, we kind of combine the two approaches, but they aren’t conducted in the way they should be. Instructional learning must take some time in the classroom, but the empirical method of getting knowledge was always valued more than just putting knowledge in the students’ heads. Independent learning is conducted at home, but it doesn’t have the same effect it would in the classroom. The same effect has the using of the ReviewEssayService.com in the classroom if it was allowed to promote the learners’ independence in developing their writing skills.
So is it obvious to anyone which voice(s) above are probably teachers of K-12 students and which are probably observers of those teachers? I, for one, think it is time for NCTM to stop suggesting to teachers what they recommend, and let teachers talk to each other about what works, and doesn’t work, day to day in the classroom.
Teachers want to turn to NCTM (an organization for teachers) to find curriculum lessons that another teacher has used, and thus done a type of research on, in his or her classroom. In reading that lesson’s progression, that teacher can then know how it might meet the needs of his or her students in learning that math, and can figure out how long that same “lesson” might take a specific set of students to learn proficiently.
To begin a conversation on how this might work, I am going to access a wonderful article from NCTM’s JRME archives: Reconstructing Mathematics Pedagogy from a Constructivist Perspective, by Martin A. Simon (March 1995). Picture a teacher looking for information about teaching the concept of area, and goes looking for that topic in a list of curriculum topics online. What help is NCTM to that teacher? NCTM currently doesn’t have such a lesson data bank organized in a way that is efficient and useful for a busy teacher. So, imagine for a second that instead of the three k-12 journals, NCTM had tabs for the three levels of curriculum – elementary, middle, and high school. Not sure which level might have lessons about area, the teacher opens the elementary school tab and looks down a list of possible curriculum big focus topics, and finding geometry, clicks on that link to open it, and then opens a link to “area and perimeter.” There the teacher finds a number of lessons that have been peer-reviewed; and with each of those lessons, additional uses and modifications of it, by other teachers, are included. Knowing that the title of the lesson is a statement of the overall goal of the lesson, the teacher opens a lesson written by Martin Simon, because the stated goal/title of the lesson is Exploration of the Multiplicative Relationship in Evaluating Area of Rectangles. (p. 122) [Note: This is obviously not the title of his article, and his lesson was actually originally given to pre-service candidates.] But let’s continue to think that this is an article a teacher to seek out at an NCTM lesson data bank.
Because these lessons are considered research documents, with empirically collected data, there would be an abstract written to introduce the path of learning and teaching that occurs in the lesson, so that teachers do not have to spend valuable time reading the lesson to know if it fits a need. Possibly it would looking like this:
Abstract – Task 1: “Determine how many rectangles, of the size and shape of the rectangle that you were given, could fit on the top surface of your table. Rectangles cannot be overlapped, cannot be cut, nor can they overlap the edges of the table. Be prepared to describe to the class how you solved this problem.” (p. 123)
Trajectory question 1: In response to students’ solutions. “Will this [method] always work? Justify your answer.” (p. 124, and not numbered in the original document.)
Trajectory question 2: “Bill said, “If the table is 13 rectangles long and 9 rectangles wide, and if I count 1, 2, 3 … 13 and then again 1, 2, 3 … 9, and then I multiply, 13 x 9, then I have counted the corner rectangle twice.” Respond to Bill’s comment.” (p. 125)
Trajectory question 3 (Optional): “Ok, isn’t it a little mysterious that we’re never counting [rectangles] here, and we wind up with a number of [rectangles]? Does that bother anybody?” (p. 126)
Task 2: (This proved to be off-target and didn’t add to the growth of understanding.) “The blob problem: How can you find the area of this figure?” (p. 128)
Trajectory question 3 (Optional): “I raised again the issue that they had brought up in solving the original problem, whether to turn the rectangle or to maintain its orientation. I demonstrated the former, rotating it 90 degrees to measure the second side.” What can you tell me about the area? (p. 128)
Task 3 (Optional): [To have students consider the usefulness of square units.] “I used your [cardboard] rectangle and my method (rotation the rectangle) to measure two rectangular regions; one was 3 x 4 and the other was 5 x 2. Draw these regions (real size). Record all that you can determine about their areas.”
In his extraordinary article (and there is an exchange that follows with Leslie Steffe), Simon defines something called hypothetical learning trajectory (p. 133), which reinforces the nature of this teaching process as a research process occurring in the classroom. In his lesson planning, he hypothesized a learning path, and then proceeded to implement it and collect formative data on it in the classroom. Recorded in the article are his active formative assessments of his students thinking during the span and progress of the lesson, and his thinking that leads to adjustments of the lesson. That ongoing assessment helped him determine the more workable path of the lesson, or its trajectory. By making these kinds of research lessons available for teachers to access and study, we begin to replicate some of the process of lesson planning that is so well known in Japan, called lesson study, (as well as the similar planning steps described by Smith and Stein in their 5 Practices books). Knowing what other teachers have done can help teachers better plan the learning progression they need, and save them time and effort.
It would therefore be beneficial to all students nationwide for these lessons to be available to the public, and especially to those who create assessments; but in order to add new lessons to the data bank of lessons, or to suggest amendments to lessons, the teacher(s) would have to be a member of NCTM. Lessons, and amendments to lessons, would need to be submitted for peer review, so that a format and quality consistency could be established and maintained over time. So I agree with all of those above who have said that it isn’t about the time, it is about the instruction required for learning.
Thanks, Matt, for raising this important issue. For students to catch up, additional time is a prerequisite. But you nailed it on the head when you state, "It is critical to note that how instructional time is used is as important as the fact that it is allocated... We need to grapple with how we organize and offer instructional time and what we do with it." The additional time is a necessary variable but not sufficient by itself.
Another time issue is time pressure. "You must understand this by Friday." "You must finish the quiz by 10 o'clock." Does it really make a difference if a student learns something a few days later, or completes a quiz a few minutes later? There are many ways to reduce time pressure, and implementing some of them can be a substantial step towards equity.
For example, what if a concept was discussed in class in Week 1, in homework in week 2, in a quiz in week 3, and in quiz corrections in week 4? That would give the student who needs it four weeks to learn it, at no cost to the student who "got it" in week 1. (That student would be pleased to move on to a new topic in week 2 class work.) Such an approach (known as "lagging homework") takes no extra time. The time is just arranged differently.
I think it is useful to highlight the emphasis on the value of informal assessment. Informal assessment is one of the nitty-gritty building blocks of successful learning because it allows the tailoring of every element of the lesson. Every question the teacher asks combined with the responses of the students, gives an instantaneous assessment and a guide on what to do next. Every question a student asks provides the necessity for an instantaneous assessment and adjustment. Even student facial expression proved informal assessment.
It is this constant assessment and adjustment that makes teaching challenging, exciting and exhausting all at once, while making it possible to make the most effective use of the mental time of everyone in the classroom.
The time question should focus more on mental-time-on-task more than on number of minutes of classroom instruction.
I agree that counting minutes is meaningless. In your example of 50 min/day vs. 60 min/day, another problem is the accumulation of 5 days of 10 extra minutes each day is equal to one day of 50 min. instruction. Also, although I agree that math and English may be more important than other subjects, I think we get on a very slippery - and dangerous in my view - slope. We all know that too often Music, Arts and PE get cut so that other academic disciplines can use that time. However, the prupose of schooling is educating whole students, and Music, Arts and PE are just as important for that purpose. So, I would rather not get into the discussion of which subject is more important than others.
Thanks for a thoughtful column.
I agree! I think the cirtical clarification Larson made was that additional support should be given during additional time that does not take away from class. I believe this includes the enrichment classes. This can be acheived with some thoughtful scheduling and collaboration.