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If all or nearly all of my students work to solve a problem the same way, it lets me know that I did a horrible job as a teacher. It means I must have shown them what to do and they are all following steps.
If I see a variety of strategies that kids can explain their understanding... wooohoooo! Does it mean I'm done? NO! I start my searches to select strategies to help students connect to efficiency. It lets me know who is ready for even more of a challenge. Challenge doesn't always mean bigger numbers or something completely different. They will dig in from a different angle. (Can you tell that I love this!!)
Why aren't we "doing math" the way we did when we were in school? Read what is in green below.
What are these wacky things my kid is writing on their paper? Scroll down to see strategy examples shown that all relate to meaning.
Pretty much... when current grownups were in school, we were taught steps to solve math problems. We were taught to memorize how different algorithms worked. We were given pages of the exact same kinds of problems to have us practice those steps until we got the answers right. If there was a word problem at the bottom of the page, we could just take the numbers and do the same exact thing that we did on the rest of the page. We didn't even have to read it!
If I just had to present steps for kids to follow to "do" math, I could already be "done" with math this year. Would it be really easy for us to just tell kids the steps? Yes. Could I just show kids math tricks? Yes. However, do kids understand what they are doing? Can they connect their understanding to new ideas? Can they pull complex problems apart and work to solve them? Do they understand when their answer is unreasonable? Are they working way too hard? Do they think the algorithm is the best way to solve all the math problems? IT ISN'T! (I'm not saying algorithms are bad. They are an organized way of getting to an answer. We work toward algorithms when kids have built understanding.)
Don't believe me? Try this: 3,000 - 2,999. You didn't use the algorithm, did you? Imagine the crossing out and putting little 1s and 0s and more crossing out and more tiny 9s.) So... this is why we are going for meaning, noticing, connecting, and problem solving. So how do you get kids to shift to this way of thinking and problem solving??
Here is probably my most used question: "What do you notice?"
Here is probably my most used question: "What do you notice?"
(And I will repeat this question a lot without saying anything else!)
(And I will repeat this question a lot without saying anything else!)
In your child's agenda, they have George Polya's problem solving, which matches the exact one in our classroom.
- Start with, "What do you know?" "What else do you know?"
- Here's a great one. "Tell me what you notice. How will that help you?"
- Ask, "What is happening? What is the action?"
- Ask, "What are you thinking?"
- Ask, "What do you have to figure out?"
- Ask, "Are you going to end with more than you started with or less? How come?"
- Ask, "How are you going to figure this out?" (If they are stuck, get them unstuck by asking, "How can you show what you know?")
** I am completely fine with your child coming to school without the correct answers to their math homework as long as they are showing their understanding and showing their thinking for each problem.
Here is the big idea for adults (including me) working with kids for math: BE COMFORTABLE WITH THEM BEING UNCOMFORTABLE. You are helping them to become problem solvers instead of memorizers. I am not saying to ignore kids. You can go back up to those questions above. You could also tell your child to open up to George, which is in their agenda.
** If all kids come to school with all the work that leads to the right answers, they are giving me the idea that they have a solid understanding. This means I should be moving faster.
** THANK YOU FOR BEING INVOLVED IN YOUR CHILD'S EDUCATION AND BEING ENGAGED IN HOMEWORK TIME. However, if you try a, "Here, let me show you how to do this," approach when working with them on their homework this could create frustration for your child. They may not be ready to understand what seems like an easy way for an adult.
** I see it as my job to understand what my kids understand. Then, we build off that. I carefully select work to share out that often builds in complexity. This way your child can make connections between what they already know and understand to the next step in becoming more efficient.
- For example, several kids are "drawing all" still for multiplication. For 6 x 7 they are drawing six circles and are making seven marks in each circle. Then they count every single mark. Is this accurate? Yes... as long as they count correctly. Do we want them to keep doing this? No, because they won't move to the next level in making connections. They are doing every problem the same way without thinking of what they already know. I call this the "automatic pilot" mode.
- So... I will select student work that shows six circles with the number 7 written in each one. Some kids will connect this to skip counting, making a table, and/or using a number line which is more efficient. (You may be thinking... my child can't count by 7s yet, right? We can break every 7 into a 5 and 2. We'll figure it out easily now!) THIS IS POWERFUL AND CAN BE APPLIED TO OTHER SITUATIONS!
- Then... I will find someone who has work that shows they know 5 x 7 = 35. Since they know five groups of 7 is 35 (they may have used the commutative property of 7 x 5 which means something different but is equivalent), they can quickly figure out six groups of 7. It's just one more 7. So 35 + 7... and someone will likely break that 7 into 5 and 2 making 35 + 5 + 2 a problem that they can solve in their head. THIS IS POWERFUL AND CAN BE APPLIED TO OTHER SITUATIONS!
- Or... someone else may "just know" 6 x 6 = 36. I will show what this means with six circles and the number 6 in each. Then, we will discuss how that visual would need to change to represent 6 x 7. (We'd need to put one more in each group.) So... how many did we need to add? THIS IS POWERFUL AND CAN BE APPLIED TO OTHER SITUATIONS!
- It's not about getting the answer for that one problem. It's about gaining number sense, flexibility, and using the understanding from this experience to make connections to what they have done, be ready to connect this to new things that come up, and work more efficiently. Yeah... I know... that's a pretty big deal. This is part of the reason that I am very passionate about this.
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