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Mathematics

- Bárbara Brizuela
- https://ase.tufts.edu/education/people/brizuela.htm
- Dean of Academic Affairs for Arts and Sciences
- Learning Trajectories in Grades K-2 Children's Understanding of Algebraic Relationships
- http://leap.terc.edu
- Tufts University

- Maria Blanton
- https://www.terc.edu/display/Staff/Maria+Blanton
- Senior Scientist
- Learning Trajectories in Grades K-2 Children's Understanding of Algebraic Relationships
- http://leap.terc.edu
- TERC

- Angela Gardiner
- Research and Design Specialist
- Learning Trajectories in Grades K-2 Children's Understanding of Algebraic Relationships
- http://leap.terc.edu
- TERC

- Katharine Sawrey
- PhD Candidate in STEM Education
- Learning Trajectories in Grades K-2 Children's Understanding of Algebraic Relationships
- http://leap.terc.edu
- Tufts University

Public Discussion

## Angela Gardiner

Co-PresenterThanks for viewing our presentation. We would love some feedback. Take a look at the following questions and let us know your thoughts:

1. What is your school doing in elementary grades to prepare students for algebra in middle grades?

2. What are the advantages of gradually introducing algebraic concepts to young children as opposed to waiting until middle grades?

3. Does the student thinking shown in this video surprise you or challenge your understanding of children’s capacity for algebraic thinking?

## Sue Doubler

Barbara and Angela,

Thanks for sharing such a clear example of a first grader making a mathematical claim. Your video example clearly brings forward her ability to represent and justify her argument with the guidance of her teacher. I’m curious about how a teacher begins to help students build the mathematical reasoning we see evidenced here?

## Angela Gardiner

Co-PresenterThanks for watching the video, Sue. You raise a great question. We spent a significant amount of time building students’ relational understanding of the equal sign, before we discussed properties, generalizing or variables with students. We started with exposure to “non-standard” forms of equations (e.g. 4= 2+ 2). We see so many students who believe that the equal sign means “give the answer” and have had little exposure to varieties of equations, so this was our starting point. We also worked with open number sentences and true and false equations before we began our work with properties of arithmetic. After a few lessons on the equal sign students began to notice the structure of equations rather than looking for “the answer” on the right side of the equal sign. When we began our work on properties with students we introduced conjecture and they began to develop their own conjectures using natural language, we did this through card games, word problems, and small group discussion. Once students were comfortable with developing conjectures we were able to introduce variable. The clip used in this video was part of a classroom teaching experiment which consisted of approximately two 30-minute lessons per week (14 lessons total) taught over a period of 8 weeks The lessons taught during the classroom teaching experiment addressed the following core algebraic concepts and practices in the 14 lessons: (1) a relational understanding of the equal sign; (2) generalizing, representing, and justifying generalizations in arithmetic (including properties of arithmetic); and (3) generalizing and representing varying, unknown quantities in algebraic expressions.

## Sue Doubler

Thanks for elaborating, Angela--very thoughtful and purposeful crafting of mathematical understanding.

## Andrew Izsak

Hi Barbara and Angela,

Have you seen students use commutativity strategically? As one example, a student who counts on might switch 2 + 5 to 5 + 2 so that there is less to count on.

## Angela Gardiner

Co-PresenterHi Andrew,

In my experience I don't see this very much in the early grades (K-2), but I do see this often in the upper elementary grades (3-5). Hopefully with earlier and deliberate exposure to the equal sign, and generalizing, representing, and justifying generalizations in arithmetic younger students will begin to develop these strategies!

## Deborah Hanuscin

What a great illustration of children's abilities! For parents watching the video, what would you want them to take away in terms of supporting their child's algebraic thinking outside of svhool

## Katharine Sawrey

Co-PresenterThanks for the question, Deborah! We believe that the heart of algebraic thinking lies in children's abilities to generalize the world around them. Everyday events can be opportunities to encourage mathematical generalizations.

For exmple, in setting the table, there is one fork for each setting at the table. Ask your child: Can you figure out how many forks we need at the table tonight? .... How do you know? ... When your child talks about "one fork for each person," it is the beginning of an algebraic-like generalization. There are many such relationships: the number of chair legs to the number of chairs, the number of ears per person. Even doubling of recipes is an entry into generalizing relationships between quantities. Encouraging children to talk about situations in terms of mathematical relationships between two quantities is one way for them to notice the general within instances of the particular.## Deborah Hanuscin

These are all really useful examples for parents! I wonder whether parents have access to information like this--

## Katharine Sawrey

Co-PresenterThanks! During this project we met with teachers in the building on a number of occasions and shared out work. The idea of meeting with parents and providing examples of scaffolding that could be done at thome to support algebraic thinking came up in those meetings and has been one angle we would seek to pursue.

## Kathy Perkins

Thanks for sharing this project! It's so important to help build these skills early, and its great that you are showing childrens' capacity to do so. I happen to have 9 year olds, and I have been so impressed with their capacity to do this sort of thinking -- luckily their school curriculum does provide some opportunities to do so.

## Katharine Sawrey

Co-PresenterThanks for watching the video and for the vote of confidence. Children really are strong mathematical thinkers and strong generalizers, we just need to give them the opportunity. That's great that your kids are exposed to this kind of thinking!

## Pam Pelletier

Thank you for sharing such a powerful example of a student talking through her reasoning. I don't have the opportunity to listen to young mathematicians explain themselves very often any more, and I will say, that she did surprise me that she could so clearly and readily share her thinking! We often underestimate students' capacity at all ages; did your teachers do that and, if so, what are a couple of strategies that teachers developed once they realized how much more students could do?

## Katharine Sawrey

Co-PresenterThanks for watching the video, Pam! Although variable notation had been

introducedin earlier activities, both the teacher-researcher and classroom teacher were surprised by how readily this student articulated the commutative property using variable notation. As shown in the video, the teacher was able to leverage the student's proposal and dig deeper into how it showed the commutative property (or, in Emma's language, how "turnaround facts" work). We went into the project looking to explore how young students might use letters to express general mathematical statements, so were planting the seeds throughout our teaching experiment, then encouraging discussion and listening for student understanding. One of the biggest strategies, "Talk to me about this," "How do you know?" Keeping conversations grounded in student understandings allows new understandings to take off!## Angela Gardiner

Co-PresenterTo add to what Katie has said, many of the teachers we have worked with have also been surprised at what their students can do mathematically. I think one of the biggest take aways for teachers, beside the questioning techniques that Katie refers to in the above response, is that we need to spend more time having rich math discussions with our students. Many teachers are tied to a 30 or 45 minute math block but have an 90 minute ELA block, these teachers promote rich writing discussions which help students build a great story, but they don't always have time for rich discussions in math. The teachers we have worked with have talked to us about making discussion a priority during their math block because they see that their students really have a lot to share, and when they share/discuss they are building their understanding! I have had the pleasure of working with young mathematicians for several years now on various early algebra projects, and listening to them explain their thinking in sophisticated ways is the best part of my job...watching their teachers' reactions to their thinking comes in a close second!

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