For some time there has been a effort to improve American students’ math performance and knowledge by pushing the teaching of algebra into the lower grade levels. Typically this has resulted in traditional algebra instruction and courses being taught in 8th or even 7th grades. This approach has met with little or no success. I would argue that it is in fact counter productive. The only way to improve algebra achievement in a formal algebra course is to lay the foundations of algebraic thinking and notation in students’ earlier school years. This book gives teachers practical and easily implemented strategies for doing that. It is clearly written and filled with case studies from classrooms that describe teacher-student interactions in detail. Any teacher of mathematics in elementary or middle school would do well to read the book and begin implementing the approaches described here. This would make a great book for a study group of teachers seeking to expand and improve their knowledge and teaching of mathematics.

Assessments such as those designed by Exemplars^® provide a practical and rigorous way of doing this. The tasks are grounded in real world experiences, include a rubric and anchor papers with which to evaluate student responses, and are specifically aligned with national standards.

I used Exemplars^® tasks as a way of focusing teachers on assessment first when working with them on implementing the PYP, (Primary Years Program), or UbD, (Understanding by Design). After selecting an Exemplars^® task suitable for evaluating students’ mastery of the central ideas or enduring understandings in a particular unit, the teachers did the assessment themselves. Then they marked their papers against the anchor papers. This led to a discussion among the teachers about the appropriateness of the particular assessment for the unit of instruction in question. When there was consensus that the assessment matched goals of the unit, the task was adopted as a summative assessment. Occasionally modifications were made to be more culturally appropriate. Sometimes the numbers in a mathematics assessment were tweaked in order to achieve more curricular alignment.

After giving the test, teachers used the anchor papers to evaluate the students’ performance. The overall results were discussed by the teachers and used to inform planning for future teaching. Often it was decided to alter the learning engagements or lessons because the Exemplars^® task was very suitable but the instruction did not prepare the students for success. Use of Exemplars^® tasks had a dual benefit. Not only was student thinking revealed in greater clarity, but also teachers were able to be more reflective about their practice.

Teachers report that their students were genuinely engaged and enjoyed working on the assessments. This differs sharply with the typical student reaction to end of unit tests. In contrast to standard testing practices that assess products, the tasks set forth by Exemplars^® provide teachers a way to examine the processes of student thinking. This is particularly important because research shows that teachers using alternative assessments focused on process are more disposed to be flexible and responsive to their students’ learning needs. (Spinelli, 2001) Teachers who routinely use process-oriented assessments tend to continue expanding their practices with a greater range of choices and strategies. (Larrivee, 2000) Teachers, staff developers, and administrators seeking ways to transform and improve instruction will find Exemplars^® an invaluable resource.

Ellen and Alan Alquist

www.learn2inquire.com

Notes

Spinelli, C. G. (2001). Interactive teaching strategies and authentic curriculum and assessment: A model for effective classroom instruction. Hong Kong Special Education Forum, 4(1), 3—12.

Larrivee, B. (2000). Transforming teaching practice: Becoming the critically reflective teacher. Reflective Practice, 1(3), 293—308.

Studying girls has given us a lot of information about this. When girls enter middle school they think they can do anything – science, math, sports. When they leave they have altered their opinion. Their sense of agency in these male dominated arenas has been dramatically diminished. The reasons for this are mostly cultural and societal. Nice girls don’t get hot and sweaty. Smart girls don’t get asked to the prom. Nerdy girls aren’t cute. This wearing down of agency same is true for minorities and other disenfranchised groups. It just happens at different times and in different ways. Why would we think that someone’s gender, race, heritage, or sexual orientation would have any impact on their success in academics or sport or anything?

We want to have citizens who believe that they do matter and that what they do will make a difference. A citizenry that believes that about itself will not be flummoxed by demagogues, will not be deterred in doing what is just for those weaker and less privileged, and will not be victims of misinformation. That is what education in a democracy is about and why successful democracies have successful educational institutions. Go forth and develop people with a high sense of agency!

Cognitive psychology has taught us many things about teaching and learning in the past 50 years. Those ideas can be seen in the advances in teaching reading, writing, mathematics, and science. However, many teachers still question the use of reading and writing workshops despite the successes that those methods have recorded. The International Reading Association and the National Council of Teachers of English have applauded those efforts and support them. School systems and teachers still question them. The National Council of Teachers of Mathematics took the lead in redesigning curriculum standards and has supported dramatic changes in the teaching and learning of mathematics. School systems and teachers still question them. The National Science Teachers Association has identified and recommended inquiry as the very essence of science and published excellent materials on how to engage students in the scientific process and how to teach them to be scientists. School systems and teachers still question them. Why would these professional organizations support changes that would undercut a child’s understanding and success?

Please join just one professional organization and read their material. I actually believe that if you joined any one of these groups and read their nine journals from one school year over and over again until you retired you would be a better teacher for it. Try it!

I wonder if we are really doing it. I would like to think that the profession is but I am not sure. As much as I believe in talking about ideas and pedagogy, I also like to see some action. There is a lot of talk about differentiation. Folks really think they are doing it. I do not see it. I see centers, a range of assignments, and performance tasks that are open ended. Are those things really differentiated or just different for different learners? How much thought went into matching the task with the learner? That is a tough question!

I see a number of excellent teachers at work but I do not see a lot of differentiation. I would like to think that I differentiate during my workshops but I continually wonder if I really do. To act, one sometimes has missteps and errors. To learn from them we must reflect on what we did, what we might have done instead, and what other results might have occurred. It is in that thinking that we will really learn to differentiate and act differently in our classrooms.