Weekly Wisdom

Weekly Wisdom

The culture of math classrooms is rapidly changing to become more inclusive. The days of math classrooms revolving around the students who understand while letting the students who don’t understand get left behind are fading into the past. THANK GOODNESS! When I was a kid usually Kevin was the only student called on in math class. Kevin was a kid who got answers in math very quickly and always correctly. Kevin was working to complete 90 math problems on a time test in 60 seconds while I couldn’t even solve them all in 3 minutes. The days of only teaching Kevin while the rest of the class strung along are dwindling!

When kids see answer getting as the way math is done they not only miss out on the connections and the value of knowing how answers are formed but they also start to count themselves out of the game. Kids who can’t get answers quickly begin to not identify as a “math person.” On this topic, enough with this I’m not a math person mentality. Once I knew a teacher who referred to herself as a not mathy math teacher. What sort of message is this sending to students? Anyways… I’ve seen kids as young as kindergarten begin to count themselves out of the math game. That is not ok. Math is so much more than arriving at answers quickly. Please listen to Phil Daro’s Answer Getting in Math for further information against an answer-getting mindset.

How are you working to shift the culture of mathematics within your classroom? Let me know in the comments below!

Changing Our Thinking: Alligators in Math Class

Changing Our Thinking: Alligators in Math Class

There are many practices from long ago that we need to rethink as teachers. This series looks to bring up those practices and offer alternate ideas that are more relevant in today’s classroom. Today we’re discussing using alligators eating bigger numbers to teach comparison.

 

The time has come in my class to begin comparing numbers using > and < but guess what? We’re not using alligators or crocodiles and there are no numbers are being eaten. Instead, the symbols are being taught in a way that students can understand. We’re using language to teach the symbols and trust me, it really isn’t too hard!

Why Alligators and Crocodiles?

As teachers we love doing things that are cute. Trust me, I love using cute things with my kids. Its fun, it makes the learning fun and it engages the kids. Cute teaching resources can come in handy. When comparing numbers it can be hard for kids to understand. Especially when we are using symbols that kids haven’t seen and symbols that look very similar.  So we did what we do best as teachers and we made up a cute story for kids to remember. The alligator eats the bigger number. It seems simple enough. The kids can get the answer and they can solve problems comparing numbers. But what if our cute teaching resources are undermining students’ understanding? Maybe not even at our grade level but at grade levels higher than us. 

What's The Problem?

When we tell kids to draw the symbol with the alligator eating the bigger number, we are only focusing on which one is larger. Every time the child draws the symbol in their head their thinking, “12 is bigger so the alligator eats the twelve” while writing 9 < 12. You’ll notice that the symbol drawn was less than but in the child’s mind they were focused on which one was greatest. Many of my students who learned the “alligator trick” will read this as 12 is greater than 9. While not incorrect, 12 is greater than 9, this should read 9 is less than 12. This is very confusing. Yes the answers are correct because they know how to draw the symbol but there is virtually no understanding of the symbol. Most kids don’t even realize that there are two different symbols!

Build Understanding

This year we began saying the words “is the same as” when we saw an equal symbol. Why not just use the word equal? Too often students understand the equal sign to mean the answer is coming next. That isn’t what the symbol means. 

The = symbol means that what is on the left is the same as what is on the right. 

Just shifting our language within the math classroom helped students gain a better understanding of the equal sign. According to the Common Core State Standards students in grade 1 should under the meaning of the equal sign. (1.OA.D.7) How many upper level students don’t realize that the symbol means that the expression on either side is equal, or the same?

We also switched up our equations so that sometimes there were answers that didn’t come after the equal sign. The equations we showed the students had the equal symbol all over from the very beginning of grade 1. Blanks were in different spots and answers were in different spots and the equal sign moved all around. This helps the kids see that equal really means the side on the left is the same as the side on the left.

This work ties in with the mathematical practice standard 6- attend to precision. Attending to precision doesn’t only mean paying attention to the problem and solving it accurately. It also is  using precise language when we talk about math. I highly recommend reading this Think Math! article about this standard. 

Use Language First

When it came time to compare two numbers this year we created three sentence strips.  We have already been using the phrase, “is the same as,” so it was not new to students. Once they had these three sentence strips, I had them build two numbers and choose which strip went in the middle. I reminded them that just like readers, mathematicians read left to right. 

To my surprise and joy they were able to start comparing numbers accurately from the very start. Some of my most struggling mathematicians found success in using the sentence strips. I was very excited to see where this was going to go. They were not only able to compare two numbers but they were also able to use the correct language while doing so! This is just what we want our kids to do! 

As we finished math class on this day, one of my students asked if he could guess what was coming next. Of course! He whispered, “I know there are symbols that go with these words. Is that what the blank space at the start of our sentence strips is for? Are we going to get to use the symbols?” 

Isn’t it a magical moment when kids predict where their learning is going and they are excited about it? Guess what, we were going to learn about the symbols next!

Then Teach Symbols... Relying on the language

After using the sentence strips to build our understanding of the language we use to compare two numbers we introduced the symbols to the class. You’ll notice that I put the symbols on post-it notes. This was done as a scaffold in case students need the symbols removed in order to practice longer without them. 

Before putting the symbols were in place, we had an understanding that in math we use symbols to represent certain words. They already knew the symbol = went with the words “is the same as” so adding two new symbols wasn’t to tricky. 

To introduce the symbols I wrote them on post its and placed them on my sentences strips. Then the class did the same. As they were drawing the symbols on to their post it notes I could hear a small buzz of observations. Many kids noticed that the symbol opens to the side that is greater and is closed on the side that is less. This isn’t a bad realization to have but it wasn’t their only knowledge of the symbols and how they work. 

As we compared numbers that day I noticed most students relying on the language and then matching the symbol with the language. They would look at a problem like 82 __ 45 and say “eighty two is greater than 45. Then they would get their sentences strips and match the strip to the problem. Finally writing the symbol on the line. 

 

What I've Learned

This is my first year as a grade 1 teacher. In the past I have taught second and third grade and my students have already come to me with the alligator story. Once you learn a trick it is hard to ever go back to reasoning. I wasn’t sure how this approach was going to go when I introduced it to my students. In fact, I waited a long time to even complete this post to see if this method of teaching symbols really even worked. Let me tell you, it did! Right now almost all of my students can compare two numbers using the symbols on their own. They now know which one means greater than and which one means less than. We still have our sentence strips to use in case we need them but the kids have eased themselves off of that scaffolding. 

 

One Last Word About Tricks

As a former Math Curriculum Leader I have a lot to say about tricks in math. So often we teach kids tricks because we think that the math is difficult and we want to make it easier. Or we learned with a trick and we really aren’t sure about the real math behind it. Math is built on reasoning. If you are able to connect knowledge and reason through a problem you can more than likely solve it. When we teach kids tricks oftentimes it eliminates reasoning. When we take away this reasoning we are limiting our students’ math potential. Mathematicians rely on reasoning more than anything else.

Next time you go to teach a math trick in your classroom can you ask yourself these questions?

-What mathematical properties does this trick rely on? If the trick doesn’t rely on any math but more of magic then it isn’t really teaching math at all.

-Is the mathematical reasoning eliminated through this trick? Math is built around reasoning and we don’t want to take this away from students. They don’t really have the right answer if they can’t reason through a justification.

-Is this trick just getting the right answer? Math is so much more than answer getting. Please check out Phil Daro on Answer Getting

 

Changing Our Thinking: Prompting Kids with “You Know This!”

Changing Our Thinking: Prompting Kids with “You Know This!”


There are many practices from long ago that we need to rethink as teachers. This series looks to bring up those practices and offer alternate ideas that are more relevant in today’s classroom. Today we’re discussing using the prompt, “you know this” when a child gets stuck.

Why Do We Prompt?

We give kids several prompts in a day. I would say most of the prompts I give students throughout the day are not academic prompts. Most of my prompts are for shoelaces that are untied, pencils that are on the floor, or behaviors that need to be changed. When we prompt students we want them to recall a very specific set of information and change something. When I say, “step out of line and tie your shoe” to a child he or she knows what to do. It helps him to recall a system of steps to go through in situations like the one he is in. Eventually, I would want to see him look down and step out of line to tie his shoe all by himself. We prompt kids to help them remember and help them create an internal dialogue for situations they might encounter in future. The goal is always independence. 

Teach, Prompt, Reinforce

Many times teachers begin prompts with the phrase “remember…” This assumes that students have already been taught the prompt. Before I can expect a student to step out of line and tie his shoe I have to show him what I mean. It might seem simple but we must explicitly teach what we mean by a prompt before it can be used by a student. During the first week of school when a child’s shoes are untied, I prompt them and show them what to do. After they catch on I can simply prompt. After a while, my words become the voice in their head. Now, when they notice a shoe is untied they simply step out of line and tie it. First I teach the prompt, then I can prompt, then I reinforce if needed. I do, we do, you do.

Using “You Know This”

If we prompt a child by saying, “come on, you know this” or “we just did this yesterday” or even “think back and remember” when they don’t know we aren’t helping them recall any information. All we are telling them is that they should know. Guess what, if they knew they would do it. If they remembered from yesterday you wouldn’t need to give the prompt today. They would just do it. This prompt doesn’t help them it only frustrates them and it frustrates you. When we prompt with our classroom we want it to guide students to the right choice and help create an internal dialogue for them.

Do This Instead

Instead of prompting with you know this, go through the steps: teach, prompt, reinforce. If a student gets stuck on a word, saying “we read that word yesterday” isn’t helpful. Instead, try this:

Teach:

The first few times teach the child the prompt. I have just selected a simple prompt and a simple situation to see what it might look like.

When I see a word I don’t know I look at the first letter and get my mouth ready to say the sound. I notice this word starts with a d. I know a d makes the sound d. Here I told the child the situation they might find themselves in- I see a word I don’t know. I next told them what to do- I look at the first letter and get my mouth ready to say the sound. Then I showed them what I meant and said the d sound. Prompts need to be short and they need to be direct. First name the situation then tell what to do.

Prompt:

Once you have gone through the teach a few times, prompt the child to have them go through the steps on their own. “When you see a word you don’t know, look at the first letter and get your mouth ready to say the sound.” This reminds the child of what they can do at an unknown word and sets them up for success. This prompts should also have the child recall all the times that you showed him how to do this as a reader. Again the prompt is short it names the situation and it tells what to do. Soon this prompt will become the voice inside their head when they approach an unknown word.

Reinforce:

After a while, the student won’t need this prompt anymore. Your prompting voice will now be the internal voice inside guiding them in this situation. They should now be able to notice what situation they are in and recall what to do. Every once in a while you will need to go back and reinforce their knowledge but they should be able to do it on their own the majority of the time. Independence is always the goal.

Changing Our Thinking

I hope this small shift in thinking is helpful in your classroom tomorrow. Try it out. Just select a certain situation it doesn’t even have to be academic and try out the teach, prompt, reinforce method.

Leave a comment below about your shift in thinking, any questions you might have, and how this is working for you within the classroom.

Let’s Talk About Answers in Math

Let’s Talk About Answers in Math


A few weeks ago I led elementary  teachers in a PD around answer getting. How many times a day in math class do we hear kids say, “But I got the right answer! Why should I have to explain it?” Think back to when you were in school. All that mattered was the answer. I remember days in high school going through math homework and all we did was go around round robin sharing the answer. If you got the answer correct it was a point, if you got the answer wrong it was no points. It has been drilled into our heads that answers are the only essential piece to math instruction and this is incorrect on many levels.

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An anchor chart now hanging in the multipurpose room to remind teachers to focus on the whole process in math.

Phil Daro one of the main authors of the Common Core State Standards in Mathematics has a phenomenal video on answer getting. Please take time to listen to the full thing. If you absolutely cannot watch the entire video here is a shortened version.

As you watch reflect on the way that you teach math. I’ll be honest there are still times when I focus on answer getting and I have to consciously stop myself.  Not all the shifts we make as teachers are easy to make but they are necessary. I created two different reflection sheets to use if you would like. Answer Getting PD

Let’s Review: Answers are important in math. I am going to repeat that so you don’t forget. ANSWERS ARE IMPORTANT IN MATH but they are not the whole story. We need to shift students’ focus to the process. There are many ways of doing this. One way that you could start tomorrow is talking about HOW students arrived at an answer.

What are some shifts you are going to make around answer getting in math? What are some math topics you would like to learn more about?