Thanks 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.

 

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

Hi 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!

Thanks 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.

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

Thanks 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! 

To 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!