Solving word problems is hard.

Really, really hard.

In a world where reading comprehension, logical thinking, math computation and visualization come together, word problems were born.

As teachers, we are interested in doing everything we can to make instruction make sense, come alive and "click" for our students. And, unfortunately, in the interest of the end goal it can be very easy to try to teach tricks to our students. In the long run, however, this does SUCH a disservice!

Let's look at the list of commonly taught "keywords" and match it up to a progression of addition and subtraction word problems from K-2 to see if learning these key words will serve a student well.

Keywords for addition often include:

**add**

**sum**

**total**

**plus**

**and**

**in all**

**altogether**

**together**

**more**

**Subtraction often sounds like:**

**difference**

**take away**

**minus**

**fewer**

**less**

**took**

**gave away**

**left over**

**difference**

*First up: Do the keywords hold up in Kindergarten?*

**CCSS.MATH.CONTENT.K.OA.A.2**

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

So, the standards alone here really aren't enough to determine

*what*word problems look like at this level. To learn about what the standards "look like" you need to look into the progressions document. Have you seen this chart? I know it's difficult to read, so I have included a link.

Commoncoretools.wordpress.com |

Back to kindergarten now. The bulk of the problems are put together, take apart, add to and take from problems where the result is the missing piece. Looking at the questions, one by one:

4 bunnies sat on the grass. 5

**more**bunnies hopped over. How many bunnies are on the grass now?

*Keyword indicates addition, addition of 4 + 5 will solve the problem.*

*10 apples were sitting on the table. I ate 4. How many are on the table now?*

*No keyword... you could argue that "ate" means take away so that's subtraction but, in the world of keywords they really offer no help here.*

*2 green apples*

**and**4 red apples are sitting on the table. How many apples are on the table?

*Keyword indicated addition, addition of 2 + 4 will solve the problem.*

*Grandma has 10 apples. How many can she put in the red vase and how many can she put in the blue vase?*

*This question is open ended and contains no keywords.*

**Conclusion?**In kindergarten, keywords are not misleading, however, they are not helpful in solving all types of word problems.

*Next up: Do the keywords hold up in 1st Grade?*__CCSS.MATH.CONTENT.1.OA.A.1__

**Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.1**

**The "1" at the end of this standard is directing you to the chart that describes the problem types. Too bad they didn't make the same note in the Kindergarten standard.... I digress...**

So here you see we have unknowns in all positions in all types of word problems. The only real caveat here is that change problems where the start is missing and comparison problems where the language is intentionally misleading are saved for second.

Let's go through a few problems:

6 bunnies were sitting on the grass.Some

**more**bunnies hopped there.Then there were 12 bunnies. How many bunnies hopped over to the first 6 bunnies?

*Keyword indicates addition. 6 + 12 does NOT solve the problem. A student needs to be quite flexible with situation and solution equations for this key word to make sense.*

*18 apples were on the table. I ate some apples. Then there were 10 apples. How many apples did I eat?*

*No keyword. If you were in the camp that said "ate" indicated subtraction before 18-10 will yield the correct answer.*

*14 apples are on the table. 7 are red and the rest are green. How many apples are green?*

*No keyword.*

Lucy has 16 apples. Julie has 9 apples. How many

**more**apples does Julie have than Lucy?*Keyword indicates addition. Addition will NOT solve this problem. Without any obvious action in this word problem, even a situation equation is a difficult argument for a 6 year old.*

Lucy has 10

**fewer**apples than Julie. Julie has 18 apples. How many apples does Lucy have?*Key word indicates subtraction. Subtraction will solve the problem... but wait until 2nd grade...*

**Conclusion?**In first grade, many problem do not have obvious keywords and those with keywords may be very misleading with a keyword leading to a situation equation rather than a solution equation. But when was the last time that you saw a "keywords" poster that said "Keywords for determining the operation in your situation equation!" I haven't seen one yet.

If I haven't yet convinced you that teaching keywords is doing a disservice to students, continue on to reading about 2nd grade.

*Last, but CERTAINLY not least: Do the keywords hold up in 2nd Grade?*

**CCSS.MATH.CONTENT.2.OA.A.1**

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1

I am not going to even TOUCH the multi-step nature of word problems at second grade. Obviously with multiple steps there may be multiple keywords for a student to sift through.

Here we add in the last 4 type of word problems.

Some bunnies were sitting on the grass. 19

**more**bunnies hopped there. Then there were 27 bunnies. How many bunnies were on the grass before?

*Keyword "more" indicates addition. 19 + 27 will not solve the problem. Addition is appropriate only in a situation equation.*

*Some apples were on the table. I*

**ate**7 apples. Then there were 45 apples. How many apples were on the table before?

*No keyword unless we count "ate". In that case, subtraction is appropriate.*

Lucy has 19

**fewer**apples than Julie. Lucy has 36 apples. How many apples does Julie have?
Keyword indicates subtraction. Subtraction will NOT solve this problem. In fact, a solution equation using subtraction would have to look like J-19=36. How many of your students would have written that? In terms of pure key words most students would write 36 - 19 as Lucy was written directly before the word "fewer". The keyword is very misleading in this problem!

Julie has 63

**more**apples than Lucy. Julie has 89 apples. How many apples does Lucy have?*Keyword indicates addition. Addition will NOT solve this problem. Most students, in an attempt to write a situation equation matching addition will write 89 + 63 = L as Julie's name was written first. The keyword, again, is very misleading.*

*In fact, in the problem type diagram for the last 2 problems, the descriptions are "fewer suggests wrong operation" and "more suggests wrong operation".*

__Overall conclusion:__In kindergarten keywords don't always help but they don't really hurt either. However, if students in K are taught key words, they are set up for trouble in first grade where key words lead to situation equations but NOT necessarily the answer of the problem. By third grade, key words can be misleading in a way which will cause students to write even a situation equation in the wrong way. *UPDATE* In participating in the book study for the book "Teaching Student-Centered Mathmatics" chapter 9 discusses developing an understanding of operations and specifically sites the problems with teaching key words. Read more about that chapter HERE.

**Suggestion?**Focus on the action of the problem, what is a part, what is a whole. Are we missing a part? Use subtraction or missing addend addition. Missing the whole? Use addition. In comparison problems, focus on the larger amount as the whole, and the parts being the smaller number and the "more/fewer" piece. If you would like to read more formal research about problem solving and the drawbacks of word problems, I have an article for you. The Elementary Math Maniac -a really fabulous blogger- sent me an article (HERE) about alternatives to key word instruction for problem solving. I thought the article was fantastic and am pleased to pass it on to you!

If your kiddos could use more problem solving practice, check out my monthly word solving packs. Students sort problems based on a missing whole or missing part or by the operation they would use to solve. Click HERE to check them out!

Amazing post! I have read lots of research over the last 10 years that shows teaching key words HURTS a students' ability to solve problems. I love how you looked at all the different problem types under the Common Core. Great analysis!

ReplyDeleteTara

The Math Maniac

I am a 1st grade teacher and we are REALLY struggling with word problems. Only about 1/2 of my students can even read all of the words in the problems.. of the 1/2 that can read it, about 1/2 of those could not read with enough fluency to gain enough meaning to identify the question or paraphrase the problem. Even when read to most of the students struggle to properly depict what goes where and when things happen. How do you suggest we get to the heart of the matter and help them identify the action of the problem?

ReplyDeleteI wonder how many students understand that putting parts together makes a whole and addition accomplishes that end. Additionally, when you start with a whole and take away a part you are left with the other part and subtraction accomplishes that end. If students don't have that understanding they are ready to fail before they start. A part/part/whole model is a beautiful way to accomplish this understanding.

DeleteOnce this understanding is in place a student can look at a model or diagram drawn from their understanding of a word problem and create an equation that will help them solve the problem.

The trick here is that students need to know how to comprehend the problem and fill out the model that you are using in your room (isn't that what we all struggle with!). The best way I have found to accomplish this end is to use hand motions so that students can "see" the action of the problem. I did a few videos on hand motions.

http://polkadotsandteachingtots.blogspot.com/2014/06/we-like-to-move-it-move-it.html *This video describes hand motions for part/part/whole problems.

http://polkadotsandteachingtots.blogspot.com/2014/07/tomatotomahto.html *This video describes hand motions for comparison word problems. The gist is that the hand motions allow students to see which amount is the larger amount, the smaller amount and the difference.

I totally agree! It makes my mind cringe when teachers talk about having students look for key words. So not helpful!

ReplyDeleteWhat I Have Learned

I teach the kids to visualize the problem and to make part-part-whole diagrams. Thanks for this post!

ReplyDeleteJan

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This is really interesting! Kids ought to be taught with clear visualisation with enhanced learning! Alin

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PLEASE continue your discussion through 5th grade! I am so against key words! In 5th grade there are so many situations where these key words actually keep the students from solving correctly. Actually, they aren't solving, because that would imply that there was actual thinking done. I'd love to share this with my school! Thanks!!!

ReplyDeleteWhat I dislike is that this is another "either/or" "good/bad" issue in education. Key words are not bad. In as much as we are solving English word problems, all words are important and useful. The point, to me, is that words help to paint a picture of something that happens in real life. If we can teach students to paint that picture in their head - visualize - the mathematical meaning will be much clearer. Sometimes key words will help, sometimes there aren't any reliable key words. To actively teach against them is, I think, a mistake. To actively keep an open conversation where we analyze problems as to how the picture is painted is our best bet. Thank you for your thoughts!

ReplyDeletePat, I was just going to say the same thing. I think it's about reading comprehension in the language of math. Key words can be a piece of the puzzle that may or may not be helpful. It's up to the reader/problem solver to discern the difference. Visualize, definitely, and don't discount all avenues available.

ReplyDeleteThanks for the analysis and starting the conversation!

Pam

Just went to a math training where we looked at this exact same chart for 4th grade for multiplication and division. Same is true. Key words are often misleading. I see Pat's point and agree that this is where discussion and critical thinking is so important. Our trainer said something that stuck. She said that if a person knows how to solve a word problem, it's not a problem; just an exercise. True problems require critical thinking and effort and perseverance... Thanks for your post. Good to think about.

ReplyDeleteI totally agree with Pat and Pam. Key words are very useful if used correctly. In the plethora of "helps" we have today, we need to use common sense and discern what the question is really asking. Most of the examples given used more, fewer, or no key words. These are not bad, but need to be coupled with "how many" which are the real key terms. One example about Lucy and Julie and apples was poorly written; it needed to be restated or omitted. Remember that helpful suggestions are just that, and should be used much thought. Most are not a "one size fits all" solution to better instruction.

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ReplyDeleteAgreed! Then when you go to 3rd grade and start multiplication, many of the words that students were taught mean you should add now mean maybe you should add or maybe you should multiply. Students need to be taught to understand the situation in the problem.

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