Tuesday, April 7, 2015

Patterns & Five Practices to Model

After having read "Orchestrating Discussion" by M.S. Smith, E.K. Hughes, R.A. Engle, and M.K. Stein, and really enjoying the idea of the five practices mentioned, I've decided to practice them for myself. Currently I'm not teaching in a classroom so all of the practices can not be applied, but I thought thinking about the practices and trying to apply them as far as I could in a given situation would be good practice and perhaps revealing. So I went and found a problem related to patterns on which to try some of them out(Sticky Triangles from NRICH: http://nrich.maths.org/88).

The five practices to model from the article are as follows:

  1. Anticipate Student Responses:  As a teacher before introducing the activity in class, do it yourself, consider and push yourself to come up with multiple ways in which it could be solved and that your students may solve it. You may consider misconceptions students might have and how that could influence their work. This allows you to be better prepared to address answers students give.
  2. Monitor Students Work & Engagement: Walk around the classroom, observe what students are saying and writing. Notice when students are nearly finished. Ask students about what they're doing and question them to think deeper or consider different aspects.
  3. Select Particular Student Work to Share: Having been monitoring their work, you have a good idea of what each student is doing and pick certain students to share based on ideas you want the class to discuss and consider and highlight perhaps different techniques used in solving.
  4. Specific Order to Student Sharing: Depending on what you want to highlight or address first, have the order students share in match this. Perhaps to help the flow of the discussion or to clear up misconceptions first.
  5. Connect Student Responses to Another & to Mathematical Ideas: What's the take away of everything? What should students be learning from this? What can they learn from another?

So 2-4 I won't be able to do, but I'll be applying and practicing 1 and 5 in relation to the problem of the Sticky Triangles. The problem shows triangles composed out of match sticks like the pictures below:

The problem asks students to explore the pattern that is occurring by finding how many matchsticks are in each, how to make the next triangle, and how to know many matchsticks to create any size triangle. 

Anticipating Student Responses:
  • All students will likely find the number of match sticks in the 3 triangles given: 3, 9, 18.
    • Finding triangles of at the next sizes, students would likely draw to figure out or if they notice the pattern quickly would calculate.
  • For triangles of any size, students then will have to find an equation or way to describe what is happening.
    • Might see the center of the second triangle as being the original first triangle. So it's 3 + 2 (3) where the first 3 is the matchsticks in the first triangle, 2 is the number of matchsticks at each extended corner and 3 for the three corners. But then the student would hopefully realize that the small triangle is not in the center of the third. So this does not work
    • Might come up with a recursive formula. You are adding a new row of triangles to the bottom of the previous. The triangles added to the bottom are the number of the size of the triangle and each triangles uses 3 matchsticks. So previous plus 3 times the size of the triangle.
    • Might look at the numbers, not the pictures, to come up with a formula. From 3 to 9 is plus 6, from 9 to 18 is plus 9.
    • Might look at the matchsticks around the edge and then the number in the center. So first is 3+0, second is 6+3, and third is 9+9. For the outside it is 3 times the size of the triangle. The inside is almost 3^s-1 where s is the size. Or again could use recursive.
    • Might notice how the number of size 1 triangles in each size are changing.
  • Likely some students will come up with formulas that apply to the first and second, but not beyond. And students who do not check to see if their formula can predict further answers.
Connecting Responses:
  • From this activity, I would hope students would take away the multiple ways in which a pattern can be described within both words and equations to represent the situation. Students hopefully can learn from others and the different techniques their classmates used to solve the problem. Understand how we can represent a pattern with an equation, and how to use variables in this case (what do they represent, where can we see it). Highlight recursive formulas and how they are represented in an equation.
  • The problem also at the end asks students to consider how their findings would change if we were composing squares in a similar manner. I might have students do this after the discussion or take home so that they can apply the ideas we talked about as a group and practice different ways they heard.

After looking at the problem in this way, I feel it really pushes me to consider and see what really can be taken away from a problem. It allowed to think about different ideas, some of which I might not have thought of, and maybe students wouldn't either. It also shows if a problem may or may not have multiple view points, if there would be anything to discuss about the problem. If a problem were fairly simple, using the five practices would reveal this. You likely would not be able to think up many different student responses. It gets you to consider your students and how they might think. I don't have students, but if I did I can see how you'd be able to think back to work they've done and how ideas they have may influence their techniques in the given situation. It also gets you to see what things should be discussed. If the problem is very computational, you can focus on different ways students think through the computations, what order do they do things in, and how can they check their work. You can get at why students are doing what they are doing, because you'll have an idea before hand and can prepare appropriate questions.

The five practices really gets you to focus in on what you are having students do, what are they doing, and why does this matter? You can better plan and be better prepared.


  1. Lovely. Good statement of the idea, and a really reasonable first shot at the practices. Some other misconceptions might be doubling from now on (b/c 9, 18, ...) and could need encouragement to draw the next one instead of just getting from the numbers. Tables would be good for organizing data here, too. Especially if you can get at why 9, why 18. If students have some quadratic experience they might notice the differences 6, 9, 12... especially if they see as adding a row of 2 triangles, 3 triangles, 4 triangles. In the connections, you might want to get students connecting representations, numeric, picture and algebraic (recursive or direct).

  2. I really enjoyed your post. A couple things stuck out, your perseverance and the idea in thinking about a problem in the way our students would think about it. Your preparation for a variety of responses was intriguing.
    The practice of connecting a variety of student responses to help them see a number of different ways to approach a problem seems the most valuable to me. Then again, if students do not come up with a number of different ways to look at a problem this may be difficult. I think encouraging students to think outside of the box would me great for improving their mathematical flexibility.