# Resources for Educators

Teachers, math coaches, and principals are always looking for better resources to strengthen their work in classrooms. We support that work by posting a changing set of useful resources that emanate from our pre-publication writings, our site-based work with students, and our professional development sessions with teachers, schools, and school districts. We also have tools and sample documents that can support your instruction. We have videos that you can study or utilize in a faculty discussion. And lastly, we have lists of recommended books and articles to help you grow your professional knowledge.

Student artifacts can serve to inspire you or can be leveraged to begin Student Work Reflection Meetings at your school. Below are artifacts we have collected from our work with students in summer sessions and at our micro-school.

This work sample comes from a first grader, after he had completed two of our deductive lessons around the concept of combing like terms. This is also, of course, addition. However, within the number lab's concept matrix, "adding" is deepened to the big idea of "combining like terms."

In this work sample you can see the child getting the hang of this idea as he combines like terms after first substituting ones for the fives. After he combines the like terms, he notates that he has 25(1). You can see that he understands the concept of like terms (the underlining is evidence of this), he understands Substitution Principle (you'd have to hear him explain his thinking to know this for sure, but he did use the word "substitution" when asked to explain to his peers), and he understands unitizing. Additionally, the clear step-by-step notation of his thinking shows that he understands how mathematicians logically note their thinking for other mathematicians.

This is work from a second grader and it is a great example of a child understanding unitizing and the multiplicative identity. If you had seen the child explain her thinking to her peers, you would have been very impressed! Her teacher actually stopped the class during studio time and brought everyone over to see what Julia had done, after she expressed these two terms as 6(5(1)). That's a common technique we use -- an interruption during partner work time -- in order to help solidify the lesson focus of the day. All the kids were asked to gather around Julia's board and observe how she'd thought about the combining of the two terms. Discussion ensued, as we worked to all notice that she had combined the like terms (i.e. four fives and two fives is six fives), but had also captured the fact that the six fives is technically six-five-ones or 6(5(1)). Julia knew she was counting fives, but also knew those fives had a multiplicative identity as well, and were actually five ones. Between Julia's notation and her clear description of her thinking, we knew this work sample showed that Julia had secured the ideas of both the multiplicative identity property and of unitizing.

Two third graders worked in partnership to simplify this expression and their work displays strong understanding of substitution property and the distributive nature of multiplication. The girls reasoned that 63 forty-eights could be thought of as the sum of 50 and 10 and 3 forty-eights. They were then able to articulate to their peers that they had substituted 50 + 10 + 3 for 63. The partners then explained they knew they could think of this as 50 forty-eights and 10 forty-eights and 3 forty-eights because she of the distributive property. Further, the language of "50 and 10 and 3 forty-eights" supported this understanding. Had we asked our typical question, "What are you counting?" Their answer would have been "48's" and the second line of their algorithm shows they are reasoning that one can count those 63 forty-eights as 50(48) and 10(48) and 3(48). Next, the girls used their knowledge that 50 is 1/2 of 100 to easily simplify 50 forty-eights and they notated this thinking beautifully! They completed the rest of the work mentally, and finalized their algorithm and asked one of the teachers, "Do you agree with our algorithm?" What a fantastic question! So much better than "Do we have the right answer?"

One of the practices that is common at our lab and a practice that teachers employ in the classrooms we work with is **Analyzing Algorithms**. Nearly every day, at all age levels, we take our students through this practice. Students are asked to look at the work of a fellow mathematician (perhaps from the same class, but often from a different class) and consider what the mathematician was thinking. Below are two examples of students working through this practice. Video#1 shows a small group of 2nd - 4th graders who have worked together for 13 days. Video#2 shows a larger group of 2nd - 5th graders who have worked together for 10 days. For more on this practice, click **here**.