Reflections from a Third-Grade Classroom

Being that I work in a first-grade classroom, I am very interested in learning about the way that children understand place value and base ten. I am planning on interviewing three students who represent the various stages of conceptual mathematical understanding. Student 1 is a young girl who is sweet, eager, and hard working, but seems to have no idea how our number system works. Student 2 is very quiet and well behaved, and seems to have an emergent level of both conceptual understanding and procedural knowledge. Student 3 is quiet but confident, and understands the procedure of math very well. I am not sure about her conceptualization, because she rarely needs help or gets things wrong.

My math CT opens class with a game that she calls “Squash.” She chooses (or more likely decides as the game goes on) a secret number between two points, draws a relevant number line, and asks the children to guess her secret number. When they guess, she plots their point on the number line, tells them if her number is greater or less than the one she chose, and shades out the eliminated, “squashed,” portion. When the children finally reach the correct number, she asks them what they can tell her about the number. What is special about it? What is it made of? Etc.

Many times during this game, I’ve heard her “reinterpret” a child’s statement so that it was correct, when the actual wording was more ambiguous. One of the most common mistakes that I see is when children explain how to break down a number, for instance, 43. Many children will say that 43 is really “four tens and one three.” My CT quickly corrects this mistake (seemingly subconsciously) by repeating, “Okay, so here we have four tens, and three ones.” Though it seems like a small mistake, these are very different ideas!

I want to see if children really understand the base-ten place value system, or if they have merely learned the “procedure” of how to respond to her request. I plan on asking the following questions:

• How does the game “Squash” work?
• How do you choose what number to pick during Squash?
• Why do you pick this number instead of that one

Afterwards, we will play one round of Squash. After each guess, I will ask:

• Why did you pick this number?
• How did you know it wasn’t going to be this other number?
• So if my number is (greater than/less than) yours, where should I shade in my number line? What have we ruled out?

After determining the correct number, I will ask the children to tell me what they can about it.

• Is there anything special about this number that you are aware of?
• Can we break this number down at all?
• How would you fill out a tens/ones chart using this number?
• (Flipping the chart so it is a ones/tens chart, and giving a new number, I will ask:) What about this one? Can you do this one too?

Finally, I will draw a pictorial representation of a number containing tens and ones (rods and units). I will finish by asking:

• What number is this?
• How did you know?
• Show me how to read this.
• (I will draw a similar picture, but not draw the ten individual units in the rod.) What about this number, how would you count this?

Hopefully, this will help me ascertain the level of understanding that my students possess. This will help me immensely as I begin planning my lessons in the near future! In a sense, this is a sample set to help me pre-assess my class as a whole.