NSF Awards: 1219605, 1219606, 0918239, 1207945
2017 (see original presentation & discussion)
Grades K-6
This video will showcase Project LEAP, a series of inter-related projects that address the overarching question of whether early algebra matters. Specifically, we address the question: Are elementary school students who experience early algebra instruction as a part of their regular curriculum better prepared for algebra in secondary school than students who only experience a business-as-usual arithmetic curriculum? The presentation will give an overview of the three LEAP projects, which collectively, provide evidence that children in elementary grades can engage in powerful ways of generalizing, representing, justifying, and reasoning with mathematical structure and relationships.
Eric Knuth
Professor
As you you view the video, you might consider the following questions:
(1) Current learning standards such as Common Core argue that algebraic thinking should be developed beginning in Kindergarten. Do you think this matters in terms of students’ success in formal algebra in the secondary grades?
(2) In what way, if any, does the information in this video challenge your thinking about the impact that the introduction of algebra in the elementary grades might have on students’ later success in algebra?
(3) In what way, if any, does this video challenge your perceptions regarding elementary grades students’ abilities to think algebraically?
(4) For our teacher viewers, what is your school doing in the elementary grades to prepare students for algebra in the middle grades?
Thanks for viewing the video about Project LEAP, and we look forward to your comments and questions.
You can learn more about Project LEAP, its early algebra curricular materials, and LEAP Professional Development, by reviewing our project brochure: Project LEAP Brochure
Wendy Smith
Associate Director
Project LEAP sounds very interesting, and clearly has been making steady progress. Can you provide some additional information about LEAP 3, in particular? What kinds of professional development did the teachers participate in prior to implementing LEAP materials? What kinds of ongoing support do teachers have access to? How did you help teachers find time to implement the LEAP materials? Do the LEAP materials replace other math units, or are teachers doing these in addition to regular district curricula? Are you using an assessment designed for project LEAP, or are you using a district or state math assessment? Does Project LEAP have any parent or community materials that teachers can use to help introduce the idea of early algebraic concepts at the elementary level to parents and other community members? Early algebra materials and other teacher supports are definitely needed to help teachers work with elementary students on early algebra concepts, so Project LEAP is definitely addressing a need in the field.
Eric Knuth
Professor
Hi Wendy,
Thanks for making the first post! :)
The teachers participating in the project receive professional development during the year (one day program overview during the summer, and approximately once a month during the academic year). The professional development focuses on reviewing implemented lessons and previewing upcoming lessons as well as on student thinking associated with the lessons. The teachers also have access to project team leaders if they have additional questions.
The LEAP materials are implemented as a replacement to the mathematics that would normally be planned for a particular day. Note that our data also show that the "loss" of the normal lessons that are replaced by LEAP lessons does not effect student achievement on state standardized assessments.
We are collecting assessment data each year using both a LEAP-designed assessment and state-mandated standardized (end of year) assessments. We designed the LEAP assessment given that state standardized assessments typically are not designed to measure early algebra learning.
We do not have any parent or community materials available, but that is an excellent idea. As we move forward with the LEAP materials, that will be something that we will plan to include.
Again, thanks for viewing and for your thoughtful questions.
Miriam Sherin
Professor, Associate Dean of Teacher Education
Such an interesting project! You make a compelling case for the need to engage elementary students with early algebra concepts. I'm wondering if you can share an example of what you mean by a comprehensive approach vs. business as usual approach - are there particular kinds of tasks or algebra concepts that you find are particularly productive for students to explore? I also think your approach to research is really important - moving from observation to research-led instruction to larger scale teacher-led instruction is essential if we want to understand what it takes for teachers to guide student learning in the ways you are finding effective. Have there been any unexpected challenges that you've faced this year, in introducing the materials to teachers?
Eric Knuth
Professor
Thanks Miriam for visiting and for your comments.
By comprehensive, we mean an approach that considers the "entire" domain of early algebra; for example, there has been a lot of prior research on specific aspects of early algebra that has provided important results about children's algebraic thinking (equal sign - Carpenter and colleagues; generalized arithmetic - Schifter and colleagues; symbolic representations - Carraher and colleagues, to name a few); however, few, if any, studies have considered the entire domain. We also view comprehensive as meaning a long term view - spanning grades rather than a single grade. By business-as-usual, we mean elementary curricula that are primarily arithmetic-focused, with perhaps some early algebra content appearing here and there.
Our project website provides some examples of tasks and lessons as well as concepts that we have found to be particularly productive in fostering children's algebraic thinking. As an example, we have found work with open number sentences (e.g., 3+4=_+5; 7=2+_) to be examples of tasks that help promote a relational view of the equal sign and that provide students experience with number sentences in non-typical formats (3+4=_ versus 8=8 or 3+4=1+_+3).
Given we are now entering the fourth and final year of our large-scale study, and are bringing our small-scale study to a close, we hope the challenges at this point are not too unexpected! :)
Our fidelity of implementation data show differences in teachers' implementation of our early algebra intervention (and subsequently, their students' performance), thus one challenge we face moving forward is trying to understand how to better support teachers in their efforts to implement our early algebra materials with fidelity.
Jeffrey Barrett
Daniel Heck
Hi, Eric and Maria.
Following up on one part of your research, namely measuring fidelity of implementation:
Can you share a bit about how you are measuring fidelity in your research?
And, based on what you have measured, what differences in teachers' implementation seem particularly important in terms of students' opportunities to learn (or students' performance)?
Eric Knuth
Professor
Hi Dan,
We are assessing fidelity using multiple measures. We utilize self-report measures, specifically teacher logs and interviews, as well as classroom observations. Teacher implementation logs are completed by teachers in our experimental group after each lesson, while regular classroom logs are completed monthly by all teachers (both experimental and control). Interviews are conducted by a researcher on our team three times a year with a subset of teachers. Observations are also conducted with a randomly selected subset of teachers, typically around 50 per year. We observe each classroom twice and code the observations on a variety of variables.
To date we have primarily used the self-report measures as descriptive data. However, we have run analyses looking at the relationship between measures of FOI within the observation and student performance on both a standardized assessment and an assessment created by the project. While analysis is currently ongoing, we are seeing some impact of teacher implementation on student performance. For example, in one analysis we found that teachers' efficiency, their handling of student difficulty, and their inclusion of student ideas was positively related to student performance on the LEAP assessment.
We are in the beginning stages of analyzing the Grade 5 data, and so will have more to say about both student performance (looking across Grades 3-5) and teacher implementation, as well as links between the two in the near future.
Miriam Sherin
Professor, Associate Dean of Teacher Education
Thanks Eric - really really interesting work! The comprehensive approach you describe is very impressive.
Jeremy Roschelle
Hi Eric and Maria,
Exciting work, and great to see some results. If a teacher or school wants to leverage your insights, what's their best next step?
jeremy
Eric Knuth
Professor
Thanks for "stopping by" Jeremy!
If a teacher or school is interested, they can learn more about Project LEAP, its early algebra curricular materials, and LEAP Professional Development, by reviewing our project brochure: Project LEAP
Gladys Krause
Research Associate
I am curious if you have seen growth in teachers' mathematics knowledge throughout your study? Also, do the teachers have any input on the content of the lessons taught?
Eric Knuth
Professor
Excellent question Gladys. We are not examining teacher knowledge as part of our research, however, anecdotally, it does seem that to some extent the PD and their implementation of the LEAP lessons are enhancing teachers' knowledge.
With regard to your second question, the short answer is yes. The lesson materials have gone through multiple revisions as we have taken into account both student and teacher experiences with the materials since Project LEAP's inception (almost 8 years ago). And in the current work (our large-scale study), teachers are expected to teach the concepts and practices targeted in each lesson, and we provide suggestions on ways in which to engage students in the lessons, however, we are not requiring that teachers follow a "lock-step" script of implementation. Much like the implementation of any curricular materials, teachers often use their professional experience and knowledge to adapt for their particular contexts.
Sybilla Beckmann
Hi, I'm curious if you have these young students express algebraic ideas with math drawings, and if so, do they also connect these to standard algebraic notation?
Also, do you have any guess about why the differences are not statistically significant at first but then become so later?
-Sybilla
Eric Knuth
Professor
Hi Sybilla,
Many students start with drawing on many of the tasks in our lesson materials and, for some students, this seems to provide the scaffolding that often leads to a generalization (expressed oftentimes with words, and then symbolically). Teachers often find that the diagrams help students to articulate a relationship in words, and expressing the relationship in words can then lead to the development of symbolic notation to represent the relationship.
Perhaps I did not provide enough explanation regarding the results: At baseline (i.e., Grade 3 pre-test) there was no statistically significant difference between the two students groups, and in each subsequent year of the intervention (end of Grade 3, end of Grade 4, and end of Grade 5), the intervention students significantly outperformed the comparison students.
Jeffrey Barrett
Eric and Maria,
Thanks for explaining your research on early algebra across all three phases. I agree with others that your work is contributing an important longitudinal component by showing the transitions that are reasonable to move from observational accounts, to researcher led intervention, and then onto teacher led instructional work in their own classrooms.
I am curious about your recommendations for other research teams or curriculum developers working to move from experimental work at small scales to implementation work at larger scales. What are key principles you expect to benefit others as they write the curricular guides and lesson descriptions for practicing teachers. What kind of advice have the teachers offered to your team after working on your project?
Thanks!
Further posting is closed as the event has ended.