Wednesday, December 28, 2011

Think About It!: The Equals Sign

Ask yourself and answer these three questions:
  • What do you understand the equals sign to represent?
  • What do you teach your students about the equals sign?
  • What do your students understand about what the equals sign represents?
8 + 3 = 11

12 - 5 = 7

33 x 10 = 330

Most people understand the equals sign to show where to put an answer. When they look at the above number sentences they will likely interpret the left-hand side as the problem and the right-hand side as the answer.

A few weeks ago, I started having my classes work on basic number sense (since they still don't have it in 4th grade--a future "Think About It! discussion!). The very first thing we talked about was the equals sign. I asked them what they knew about it and all of them told me something along the lines of "it means the answer" or "it means 'is'; where you put the answer." I was a little taken aback and suddenly realized why they struggle with very basic problems such as 12 + __ = 20 or 7 x __ = 140. (Do your students struggle with those too?)

I have seen several recent studies that involve asking students the same questions about the equals sign. Let's, for instance, use 4 + 7 = __ + 2. On problems like these students would oftentimes add 4 and 7 and place the sum in the blank so it reads 4 + 7 = 11 + 2, completely ignoring the 2. I'm sure you as an adult (and teacher) look at that and instantly understand how the number sentence and thinking is wrong. Unfortunately, most students will not.

The equals sign indicates equality. It is placed between two things of equal value. Think of it as a balance. So when you say 8 + 3 = 11, you are saying that 8 + 3 has the same value as 11.

Can you see how knowing this in elementary school will help students better understand algebraic equations?? When students are learning about evaluating algebraic equations such as 3x + 7 = 28 they MUST understand that both sides are equal or they will never understand how to solve for x. The time to teach this is NOT in middle school but in the early grades.  

Students should also understand that one side of the equal sign does not always have to have a single number. The equals sign can show two expressions that have the same value, like 9 x 4 = 12 x 3.

I want to address one more misconception--the running equals sign. Here is a word problem and example:

Sally had 4 marbles. She cleaned her room and found 3 more marbles. Her friend then gave her 8 marbles. How many marbles does Sally have now?
Number sentence: 4 + 3 = 7 + 8 = 15

How is that wrong? Remember, both sides have to be equal. The correct way to write it would either include multiple number sentences or parentheses:

(4 + 3) + 8 = 15

OR

4 + 3 = 7
7 + 8 = 15

I wouldn't say repairing this misconception is a quick fix, but it is definitely doable. (And, of course, it would be better if instruction about the equals sign is correct from the beginning.)
  • You can start by having a discussion about it with your kiddos. Bring in a balance to help you explain (for instance, put two pennies on one side, then add 6 more to it. Record that as 2 + 6. Then add pennies to the other side (8) and let your students witness how the trays become balanced. Record it as 2 + 6 = 8.)
  • Another activity would be to have students use equality cards. You can make some simply by putting a number sentence with one missing number on one card and have the students match it to a number sentence in the same fact family with a different number missing (e.g., 6 + __ = 8 on one card and 8 = __ + 2 on the other.) Look for a free Equality Card activity in my Teachers Notebook store!
  • You could also make a mat with an equals sign and have students place cards of equal value on either side of the symbol. The cards could contain pictorial models, standard form, expanded form, etc.
Sidenote: I'm not sure if you are familiar with the National Library of Virtual Manipulatives--if you aren't, you NEED to be! It is exactly what it states. The site has every manipulative imaginable in virtual form for FREE. It also has activities to go along with each.

To those that teach the higher grades--check out this balance that provides visual representation of algebraic equations:

Awesome, right?!?

Even if you don't teach the higher grades I encourage you to check out the algebraic equation scale and ponder about how easy it would be for our current students who do not understand an equals sign.

Think about it...

I always enjoy incorporating literature into math class. I have not actually read this book but it looks like a cute little story about creating equal sides. Check it out by clicking on the picture.




Thanks for tuning in. I WOULD LOVE TO HEAR YOUR THOUGHTS! THROW 'EM AT ME! :D


Also, I would LOVE to tell you to look for these posts on a certain day and time but I can't right now--life is not that predictable! Installment number 2 will be before school starts back though! :)

29 comments:

  1. Great post! I love to see some true reflection going on. I start out the year with my Kindergarteners teaching them that equals means "the same as." My feelings about math changed when I was trained to use Cognitively Guided Instruction. It completely opened my world and the way I teach. I can't wait to read what you will reflect on next!

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  2. I LOVE LOVE LOVE this post! Thanks for making me "Think about it!" I can't wait to read all your other Think about it posts!

    ~Lorraine
    Fabulous 4th Grade Froggies

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  3. Wow. That really makes me think about the equals sign. I have to say that I've talked about it, but never thought about going so in depth. I'll have to work on this with my firsties. It would be good to help understand 2+2=3+1 even now. Thanks. I'll be tuning in again.
     Chrissy

    First Grade Found Me

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  4. Wonderful post! I work with all age students and am guessing that there are other common times when students say the word but miss the point. This is a biggie. Thanks for posting
    - Lisa
    a teachers bag of tricks

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  5. This is a great series which I hope you continue. Thank you for getting me to start thinking how to be more explicit in my teaching.

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  6. Not sure how old you are, but as a kid we had this game where we would put numbers on one side and the other side had different numbers. Ifnthey wer of the same value, then the balance would be level. There is great merit in everyone teaching this to their students and modeling a balance scale to show this process.

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  7. Here! Here! We do our kids such a disservice by teaching them incorrectly on this simple matter. I took a math workshop a couple summers ago that highlighted this misconception. Love this post. Keep it up!
    2B Honey Bunch

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  8. Great post!!! I teach 4th & 5th grade Math and am looking forward to more posts! Thank you.

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  9. I'm so glad you're addressing these issues. Number sense is a huge issue with our kids, and I think a lot of it has to do with the drive toward testing at such early grades. Kids aren't given enough opportunity to manipulate numbers and values within a structure and see what happens!

    There are ways we can integrate these kinds of experiences into kids' math instruction starting at an early age. If we can do this, kids will actually understand what they're doing when they are performing operations!

    Keep up the great posting!

    Buzzing with Ms. B

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  10. Love this! I actually had this discussion a couple of weeks ago - my 9 year old is gifted mathmatically and he understood that the equals sign means "the same as" but my 12 year old daughter had no real concept of what the equals sign meant! My district is actively working on this, but there solution is to just ban the word "equals" altogether. Now we're supposed to call it "the same as sign" so an addition sentence would be read as "4 added to 3 is the same as 7" for "4+3=7". Not a bad concept entirely, but I have a problem with not teaching "plus" "minus" and "equals" because that is what the rest of the country is still going to say. So I think we do a disservice by not teaching the CONCEPT instead of just replacing vocabulary.

    By the way, I have a little monkey scale from Lakeshore and you hang bananas on either end. Thus children can see that 2 and 3 bananas on one side is the same as hanging the 5 bananas on the other side. Kind of hard to explain it, but it works and it's just cute enough to motivate the students to want to explore it!

    Can't wait to read more, good job!

    Jennifer @ Herding Kats In Kindergarten

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  11. We use Bridges math in my district {not sure if you're familiar with it} but that is a concept I taught for the first time using this new math resource...= means "the same as". My firsties were thrilled with this piece of information and it really made sense to them...

    Great post!

    Holly
    Crisscross Applesauce in First Grade

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  12. Great post! I am a member of a math lesson study that includes teachers from grades K/1 (ME!) to 8th grade. We discussed this concept this year and so I have changed my way of doing "Number of the Day". Number of the Day is when I give my kiddos a number like 10 and they have to think of equations that will equal ten - for example, 8+2=10. Now I list some of the equations in this format: 10=2+8 so that the children get used to the equals sign meaning "the same as" rather than "the answer is coming now".

    I love math and am so glad you are taking the time to reflect on it! This is so good for all of us! Thank you!
    Camille
    An Open Door

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  13. Did you read my mind?! After observing the mid year state test my students took in December, it is SO important for [3rd grade] students to understand this concept. I've been thinking about how to present the concept, and the balance is perfect. I never thought, until recently, to explicitly teach the concept of a "balanced" equation in elementary school. Thanks for you post, and I can't wait fo more!


    Meg
    Third Grade in the First State

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  14. I LOVED this post and can't wait to read more like this!! And I really loved your freebie from TN--thank you so much for sharing all of this Janaye!
    Kristen
    Ladybug's Teacher Files

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  15. I totally agree! I teach 5th grade and I posed this problem a few weeks ago: 5 + x = 12 + 8

    The most common response was: 7. I guess they just ignored the 8? They were thinking 5 + 7 =12

    The second most common response was: "That's not right. Why are there two numbers on both sides of the equation?" That's when I realized that this error was a mistake on my part. We've got to show students multiple representations of content knowledge.

    Oh, some students were correct with their answer, which made feel a little better about myself.

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  16. Yes!! The Balancing Act equations are THE hardest...why?! lol...and also - another "equal" that's difficult, for my 2nd graders especially, is the line that you draw under a vertical math problem...I don't know why, but they DON'T want to put that when writing out their math problems...
    Jen ROss
    The Teachers’ Cauldron

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  17. When I started teaching 4th grade 7 years ago, I ran into the same thing you did with your students. Both districts I have worked in use inquiry based math, and we have taught more of the why behind math than I ever learned growing up. Through the years, less of my students come to my class thinking that it means the answer comes next, and most understand that it means the two sides are equal. It is great to see this change, and I do attribute it to the way we are teaching math. The algebra that I am able to do with my class just because they understand the concept of the two sides being equal is well beyond what other fourth graders are doing. It is so much fun!
    I love math, and it is so much fun to teach now that I understand how kids learn math concepts. I teach in Utah, where we have an Elementary Math Endorsement, which makes us "math specialists." It was such an eye opening experience to learn about the research on how kids learn math differently and at different times, just like they learn to read. I love that you have started this conversation, and look forward to reading some more of your ideas!

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  18. Loved this post! Sometimes I think the sad truth is that many teachers are uncomfortable with math themselves! So the kids walk away at times not knowing the basics and whys of math. Thanks for starting the conversation. I just love it.
    Adrianne
    www.kidscogworks.com

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  19. As a 7th grade math teacher I deal with the misconception of the running equal sign. I have seen this misconception perpetuated by teachers (and I have probably been guilty of it myself) while teaching order of operations.

    It is amazing how hard it is to break these types of misconceptions that my students have by the time they get to middle school.

    Great post!

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  20. I loved this post. Thanks so much for your hard work.

    Tanya
    A Class Act

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  21. I LOVE THIS POST!! I have been reading your blog for a while (and bought a few AWESOME activities from your store)! Last year I was shocked at how difficult it was to do checks for algebraic equations because the students had NO IDEA that = meant that both sides were the same. We are still working on it now, even with my smarty Algebra kids!

    Kate
    http://tothesquareinch.wordpress.com

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  22. We talk a lot about "balance" in my class. Math, like many things in life, is all about balance. If you don't have balance in an equation (the two sides don't equal each other), math won't work. I use concept lessons and "Two Problems" a lot, which help with this idea of balance.

    On a related note, have you used Hands-On Equations? It is perfect for teaching this idea of balance and equality.

    I look forward to the the next installment.
    ~Stephanie
    Teaching in Room 6

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  23. As a second grade teacher, I have this conversation with my students when I teach finding the missing values for number sentences such as 50+10= __+40. They automatically want to put 60 in the blank. We have to go through the motions and use the vocabulary over and over again. Some get it, some don't. But I did not realize the misconception until I started teaching this concept in 2nd grade. Thanks for sharing your thoughts!

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  25. Great post! I really enjoyed reading your explanation. After 2 decades in 1st grade before moving to 4th 4 years ago, I find it helpful to move from the concrete to abstract with 4th graders too. When teaching about equality, I talk about how a see-saw works. The students always discuss without my prompting them how you have to get both sides the same or someone always stays up in the air. (If your school has a see-saw, I'd recommend starting your lesson there.) The analogy of balancing people on a see-saw really helps some students understand that both sides of a math equation have to balance too. After the discussion about making both sides the same or equal, we use scales for balance equations with objects and then move on to equations. (Of course, some students don't need all of this so you can move on to the actual written equations.) My students are much better at thinking of the equal sign as showing balance or equality and not "it means the answer comes next" thanks to the visual image of a see-saw.

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  26. I do teach that -whatever is on the left side of the equals sign must have the same value as what is on the right side, and am amazed to see 3rd graders actually try to FIGURE OUT the answer to (ex) 4+3=3+ __ . I like the idea of using pennies on a balance scale to get this across conceptually. In Washington, 3+4=7+5=12 is called a run-on equation. If you're asked to write an equation to show the solution to a problem and use a run-on equation, you will not earn the highest possible score even if the answer is correct. Thanks for pointing out/reminding us to review this with our kids.

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  27. Great post. I know exactly what you mean about 4th graders and number sense. I spend a great deal of time working on this in my 4th grade class. I cannot wait to read your next post. You always have great ideas.

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  28. I'm back. I just wanted to share with your some of the learning from our class this week. I even had one of my kinders say the equal sign means equality! :) Can't wait for your next Think about It Series post.
    http://learningwithmrsparker.blogspot.com/2012/01/balancing-act.html

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  29. I'm an intern in a first grade classroom, and we are strting to explain to the kids that "equals" means "the same as". Thank you for this post and great ideas, including the book Equal Shmequal!

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