Prepared by Peter Liljedahl for the Coquitlam School Board

Mathematics 8/9 Articulation:

Middle School to Secondary School

The content of this report is based on a series of six meetings with a specific Middle Schools and a specific Secondary Schools. Based on the conversations at these six meetings I have been able to make some conclusions, and subsequently, some recommendations regarding the difficulty experienced by students as they transition from Mathematics 8 to Mathematics 9. Although the basis for these recommendations stem from the conversations with these two specific schools, they may still be germane to other school pairs. Thus, in reading this it would be useful if you try to read yourself into the report.

The Problem

There are high failure rates among Math 9 students. The feeling is that these failure rates are due, in part, to the transition between Math 8 and Math 9. One prominent idea as to the reason that this transition is so problematic is that students are not being adequately prepared for Math 9 while they are in Math 8. This lack of preparation is centred on three issues: curriculum, rigor, and assessment. The feeling is that Math 8 teachers may not be delivering the entire curriculum to a necessary depth of instruction, not exposing students to the rigor required to succeed in mathematics, and not using assessment instruments that accurately measure a students actual ability. As a result, grade 8 students are coming out of Math 8 unprepared for the demands of Math 9. At the same time, these grade 8 students have 'inflated' marks that cause the parents of these students to have unrealistic expectations of success in Math 9.

Possible Systemic Causes

However, the assumption that high failure rates among Math 9 students are only caused by a lack of preparation would be narrow sighted. There are a number of systemic causes that need to be considered as well.

In the Coquitlam School District the transition from Math 8 to Math 9 coincides with a larger transition from middle school to high school. The effects of this larger transition are not to be ignored. Research has shown that the process of changing schools has an affect on student achievement. See for example:

Neild, R. & Weiss, C. (1999). The Philadelphia Education Longitudinal Study (PELS): Report on the Transition to High School in the School District of Philadelphia.
Alspaugh, J. (1998). Achievement Loss Associated with the Transition to Middle School and High School. Journal of Educational Research, v92 n1 p20-25 Sep-Oct 1998.
Pantleo, S. (1992). Program To Reduce Failure Rates of Ninth Grade Students. (this document accompanies this report as an attachment).

The transition from Math 8 to Math 9 coincides with a transition from concrete mathematics to abstract mathematics. Up until, and including, Math 8 most of the concepts taught are anchored in real world applications. It is clear to students where the things they are learning can be applied; the mathematics is real. Beginning in Math 9 most of the concepts taught are abstract, with very little transparent applications to the real world. This too is a transition that is not to be ignored. Reports from around the province show that failure rates in Math 9 are high even in districts not using a middle school model. Research shows that the marked change in the perceived purpose of mathematics is problematic for students.

Schoenfeld, A. (1985). Students' Beliefs About Mathematics and Their Effects on Mathematical Performance: A Questionnaire Analysis.
Shannon, G. & Bylsma, P. (2002). Addressing the Achievement Gap: A Challenge for Washington State Educators. Washington Office of the State Superintendent of Public Instruction, Olympia.

Related to bullet number 2 (above) is the marked difference in the purposes for teaching mathematics in Math K-8 versus Math 9-12. The teaching of Mathematics at any level has a number of purposes: teaching to think mathematically, creation of numerate students, preparation for the next mathematics course, etc. In teaching Math K-8 these different focuses can be summarized as: preparing students to be numerate and preparing students for the next math course. To ensure that students are prepared for the demands of both further education and the workplace, the early years of the mathematics curriculum (K to 7) must help students develop mathematical literacy (BC IRP's Rationale). However, in Math 9-12 the tendency for teachers is to focus on only on preparing students for the next mathematics course. This further ingrains the attitude that there is no real purpose for mathematics in the minds of the students.
The transition to high school coincides with a transition into a semestered school. As a result, half of the students entering grade 9 will not see math until the second semester. This extended absence from the learning of mathematics likely has an impact on achievement.

Observations

Middle School

The teachers and administration at the Middle School are clearly dedicated to the teaching of students. That is, their main focus is on the student. They take great pride in the fact that they work with a mastery level philosophy (they work with the students at the level that the students are at, and they move on when the students are ready). Sometimes this philosophy prevents them from 'getting through' the entire curriculum. The teachers work in teams that are organized according to 'communities' of students. That is, they are student based and not subject based. As a result:

There seems to be very little, if any, dialogue among grade 8 teachers (and administrators) around the subject of Mathematics.

There is no scheduled meeting times among the staff which are dedicated to the discussion of Mathematics.
There are no school wide standardized assessments in Mathematics.

In summary, the teachers at this Middle School are, for the most part, covering the prescribed curriculum and are using assessment practices commensurate with the middle school model of instruction. There are no initiatives in place to ensure the delivery of a consistent or comprehensive curriculum. However, they are open to recommendations for such initiatives.

Secondary School

The teachers and administrators at the Secondary School are clearly dedicated to the teaching of Mathematics. They see this as an important subject and as such have put tremendous effort into doing a good job of teaching it. They seem to have regular and ongoing dialogue around the subject of Mathematics in general and the transitions from year to year in particular. They have created two locally developed courses to help with the transition from Math 8 to Math 9. The first of these is an Intro Math 9 course meant to review the prerequisite knowledge needed to succeed in Math 9. The other of these locally developed courses is an Essentials Math 9 course meant to prepare students for entry into the Essentials Math 10 course. In teaching the mainstream Math 9 course the teachers spend a considerable amount of time reviewing material from Math 8 so as to prepare the students for the Math 9 curriculum. Even though they spend this time doing review they are driven to complete the curriculum so as to best prepare the students for Math 10. As a result:

There is excellent communication among staff.
There is consistency of content covered.
The only purpose for teaching a given mathematics concept seems to be in order to prepare the students for the next grade.

In summary, the teachers at the Secondary School are covering the prescribed curriculum with modifications implemented in order to better prepare students for transitions from grade to grade. There are initiatives in place to ensure the delivery of consistent and comprehensive curriculum. There is a great concern over the Math 9 failure rates, but a resistance to looking towards themselves as being the source of either the cause or the solution to these failure rates. This is problematic in that the failure rates at the Math 9 level are not inconsistent with the failure rates in Math 10 and Math 11.

Conclusions

There is a problem regarding the transition into Math 9 and student achievement. Although the obvious root to this problem lies in the delivery of the Math 8 curriculum, this is not the only root (in fact, it may not even be the most significant root). As such, focusing efforts only in the delivery of the Math 8 curriculum, while alleviating some of the problem, will likely not eliminate the entire problem. To solve this problem a cooperative effort is required that will require both schools to examine and coordinate not only their curricular objectives but also their philosophies regarding teaching of mathematics and the management of students in general. More than anything, this will require critical self-examination of teaching and assessment practices.

Recommendations

To help achieve the solution outlined in the conclusions above I present, here, a series of recommendations.

General Recommendations

There needs to be attention paid to the transition from middle school to high school in general (see attached document).
Mathematics needs to be delivered in such a way as to create a seamless transition between Math 8 and Math 9. This will require a ramping up of the delivery method on the part of the Math 8 teachers and a ramping down of the delivery method in the part of the Math 9 teachers. This ramping can be gradual. While the beginning of the Math 8 year may look very different from the end of the Math 9 year, the goal is to have the end of the Math 8 year look very much like the beginning of the Math 9 year.
Students need to come out of Math 8 with greater understanding (not just 'skills') of fractions, decimals, and integers.

Specific Recommendations: Middle School

Find time to meet around the topic of Mathematics. Use this time to formulate a way to ensure the delivery of a consistent and comprehensive curriculum.
Put a greater focus on ensuring that students have a good understanding of fractions, decimals, and integers. This can be best achieved if these topics are covered early in the year so that they can be revisited in the context of subsequent topics such as algebra, geometry, and data management. Also, review the concepts from fractions, decimals, and integers repeatedly through the year.
Come together as a staff to produce and administer a cross-grade exam two times during the year (one in December/January and one in June). These exams should focus on, but not be limited to, the concepts from fractions, decimals, and integers. There should be questions that measure skills as well as questions that measure understanding. The purpose of these exams is twofold: to create a focus point for the discussion of mathematics curriculum and teaching (see the first bullet above) that will form the basis of a consistent and comprehensive curriculum, and to ramp up and acclimatize the students to more intense forms of assessment. I suggest that the results of these tests be reported out to parents separately so that they can begin to get an accurate picture of their child's test writing performance. I make no recommendations as to how the marks are to be aggregated with the rest of their course work to formulate their Math 8 mark.
The results of the aforementioned exams are to be used by the teachers as feedback on their own ability to deliver the curriculum.
Find ways to have students demonstrate what they know and understand on a daily basis (note: this is different from having them demonstrate what they can do). This will require them to do some writing.

Specific Recommendations: Secondary School

Revisit the reasons for why mathematics is taught, in general, and why specific topics are taught, in particular. That is, find ways to explain to students why what they are learning is relevant and important to them. The reason for teaching mathematics needs to be something other than, "we teach Math 9 to prepare you for Math 10". This does not mean that everything needs to be applicable to real life. There are many reasons for teaching something: it has a rich history, it has an important use in real life, it is interesting, and it leads to something in the future that is rich, important, or interesting. These reasons need to be shared with the students.
Incorporate assessment strategies other than test, quizzes, and homework. This is especially important at the beginning of the Math 9 semester. Assess for understanding and not just skills through the use of problem solving assignments (note: this does not mean word problems. See attached Math 8 Numeracy Assessment for example).
Find ways to have students demonstrate what they know and understand on a daily basis (note: this is different from having them demonstrate what they can do). This will require them to do some writing.
One of the best indicators of effective teaching at the Math 9 level is an affective measurement instrument. If students do not like mathematics, do not feel they are capable of doing it, etc. they will not achieve in mathematics. Consider using such assessment items on every test given in Math 9.