If Sameera, Bobby, and Jose buy ten Snickers bars and then chase their dogs around the neighborhood for seven minutes, how many donuts will the monkey need to bring to the zoo in order to celebrate Javi’s birthday?
Raise your hand if this seems totally normal for a math class…
Right. Well… this “word problem” pays homage to that joke you’ve probably seen on Facebook mocking every math class ever. But I honestly think that struggling learners really do feel this way. They look at the words and their heads start to spin. What does Javi’s birthday have anything to do with chasing dogs?
EYES (robot voice): INPUT. PROBLEM.
BRAIN (robot voice): OUTPUT. NONSENSE.
It’s our challenge as math teachers to help students develop the skills required for decoding the “gibberish” and making sense of it. The phrases and meanings that come so naturally to us (like “is” for =, “per” for *, etc.) are not so natural to the students. We seriously need to review these concepts each year, if not bail ship and turn to direct instruction. The Catch-22 is that merely giving a definition will only serve to go in one ear and out the other. Sum, Difference, Product, and Quotient? Forget remembering (Ba dum tsss!), much less being able to write out a phrase correctly using one of those four.
The kids have to truly grapple with key words, and revisit them over and over again until shouting “division!” is their automatic reflex when “quo…” even drifts into their subconscious.
Operations Word Sort:
Early in Algebra Part 1, I use an “operation words” card sort in the simplifying expressions unit.
- With an expo marker I write one of the four operations ( +, -, *, / ) on the inside corner of each desk in a group of four.
- Each group works together to cut the card sort (the more students can do, the better)
- They then sort the vocabulary words into the appropriate categories.
- Pictured below are a my table set up and part of the word list I use:
We then regroup for a whole-class discussion. I call on representatives from each group to share one word and its meaning. As we discuss the placement of each word, students record the vocabulary words in the appropriate categories on their individual note sheets. By the end of the discussion all students have contributed to the conversation.
Update 5/13/20: You can find a digital version of the card sort on Teachers Pay Teachers HERE.
Context Word Sort:
Fast-forward to a few months later when we begin solving equations…
The students not only have to understand operation words like sum, less than, and times, but they also have to understand the contextual implications of math words like “per,” “for every,” “initial fee,” etc. So we do a few days of practice with UPSC (to be explained in about 2 seconds…), and then I lead students through a new type of word sort.
This time they each complete their own cut and paste with the old (read: operation) words AND the new (read: context) words combined. I *love* this activity because one can argue that words like “half” might be classified as division or multiplication … or even as subtraction if you know the exact number of items you started with. Our awesome conversations lead to more complex thinking and help students to begin embracing the ambiguity of word problems.
Update 5/13/20: You can find the link to both a paper and a digital version of this card sort on TPT HERE.
And now… what you’ve been waiting for…
HOW I TORTURE MY STUDENTS WITH WORD PROBLEMS (AKA UPSC): A FASCINATING EXPOSE BY MISS ELSIE
Starting Point: Have a student read the question aloud – preferably one whose name appears in the problem….
UPSC: The Acronym
U – UNDERSTAND. This is a 2-part question:
1) What is the question asking me to do?I emphasize the fact that the question or the “task” is usually presented in the last sentence or two of the word problem. It usually entails something like “write and solve an equation to determine the number of…” or “find the number of…”
At first you can even have students copy the question’s instructions word-for-word into that space! Then with additional practice guide them into paraphrasing.
2) What do I know? Read through the question and underline the key numbers *along with their corresponding nouns*. (ESOL students tend to write down just the number 9, rather than writing “9 dogs.”)
When introducing the UPSC scaffold, use different colors to represent the 2 different parts of the question. Have students underline the question with one color and the given facts/information with the other.
P – PLAN.
- Because we’re practicing writing equations, I focus on decoding the vocabulary with my students.
- We identify our variable in the question and choose a letter to represent that unknown. (Ex: “Use c for cookies” or “c = # of cookies”)
- We also write down the operation & application words and their corresponding symbols. Ex:
- = for “total” (or… “is,” “equals,” “is equal to,” “final amount”, etc.)
- * for “per” (or… “for every,” “each,” etc.)
- + for “in addition to” (or… “and,” “earns,” “grows,” etc.)
- > for “no less than”
- The symbols students choose for certain words will determine how they write their equations. For example, in the context of a person going to the store to buy fruit, should we use addition or subtraction? It all depends on if we’re counting quantities of fruit or subtracting our costs from a total dollar amount. It’s all up to the kids and their master plans!
- If you choose to have students work with other graphical or pictorial representations, you can have them write a plan to “create a table (or draw a graph) that represents the number of hours worked and the total money earned,” etc.
S – SOLVE.
- The “Estimate” line encourages students practice their estimation skills. Often, for struggling learners this is truly a guess. However, their estimates will become more intuitive with practice. They *love* getting the answer right on the first try.
- Next, my students write their equations using the plan they just created. If you encourage students to include other representations, those would fit here as well.
- My students then solve their equations using the river method. (You can find an in-depth explanation of that approach here.)
C – CHECK.
- The students need to make sure their answers make sense!
- Students can write a sentence that incorporates the answer to their equation, showing that they’ve addressed the question at hand. This also really helps ESOL students to practice with different verb tenses and reformatting questions into complete sentences.
- Students can also substitute the answer back into the equation to show that it does, in fact, make the equation true.
While I model the first few examples of UPSC step-by-step, eventually I ask students to think about the U before we discuss. And then they contemplate the U and P sections before we debrief as a class. Finally, my team teacher and I ask students to check their equations with us before they begin to solve. In the long run, as students become more comfortable with decoding word problems, you can slowly remove the UPSC scaffolds because they no longer need the extra support. They can always write out their own UPSC frame as they feel compelled to do so.
In a different vein, you can absolutely re-introduce UPSC every time you explore new application questions and topics with your Algebra 1 students – with writing and solving systems of equations, solving quadratic equations, and the like. I guess I hadn’t thought that far ahead because Algebra Part 1 does not incorporate those topics….
I won’t lie. About 2 things:
1) I can’t take credit for this process. I have to give a shout-out to BVD, DM, MP, and a few others for all of the work they put into the research, drafts, and templates we created. But all of that collaboration led to something awesome.
*Side Note: The year we implemented this model, our team’s Virginia SOL scores jumped from a 52% pass rate to a 74% pass rate. Though the river method of solving equations was probably the greatest contributor to our success, our consistent implementation of UPSC also contributed significantly to students’ mathematical reasoning.
2) I was NOT excited to introduce this method of analyzing word problems. It requires a lot of space for one question. It’s clunky because students are literally copying down information and writing sentences. And math is about solving equations and doing MATH… right?? But really, that’s the beauty of it. As students underline and write down the key phrases, indicating the question and the givens, they’re processing the information they otherwise struggle to access. UPS(C) *delivers* (if you will… Ba dum tsss!) a tool and a safety net that helps students to embrace the ambiguity of it all. By adding structure, we remove the fear factor and give students a fighting chance to at least start interpreting what they’re reading.
Since that first fateful day 4 years ago, I have implemented and modified the UPSC model quite successfully in my ESOL classes. I’ve gone to the extremes… reducing the UPSC questions to merely “write an equation” … AND expanding them to include multiple “Solve” sections for multi-part questions about linear functions. I actually believe so strongly in its success that I’m using it as a data sample for my teacher evaluation. What can I say? I live dangerously. True to form, students scored 0, 1, or 2 points out of 10 on the UPSC pre-assessment. However, 95% of my ESOL students earned at least 6 out of 10 points on the Mid-Year assessment in January. By the end of the year I expect most of my class to attain a 7 or higher on a multi-step linear UPSC question. And that, my friends, will be a major accomplishment.
I just attended a staff development yesterday on “CueThink.” I was like, “NO WAY! THIS IS PRACTICALLY WHAT WE’RE DOING ON PAPER!” They ask slightly different versions of the same questions we pose to our students, with the added benefit of allowing students to videotape their explanations and interact with others’ work. The differences:
- CueThink asks “What do you notice?” and “What do you wonder?” …Similar to our “What is it asking?” and “What do I know?” questions, but definitely more big picture or existential, if you will.
- As a side-bar, I nixed the “what do you wonder?” prompts for high school Algebra word problems because I felt that it would only lead to smart alec responses like: “Why do I have to do this?” and “How hard to I have to press for my pencil lead to break?” High school students have *such* a hard time relating to the equations they’re supposed to solve on a computer-generated test like the SOL. Yes, we can make word problems more engaging for our own classes, but that’s an entirely different topic for another day.
- In IB Math I modified these prompts for our group dissections of challenging trig application questions. I kept asking “What do you know?” and “What do you need to know?” in order to answer a particular question. This led to some great discussions and helped to model an effective process for what students might do when they hit a metaphorical wall.
- CueThink provides different strategy choices/buttons in the “Plan” phase – draw a picture, make a graph, make a table, etc. It doesn’t focus quite so much on the vocabulary break-down, but I think that’s been my emphasis because we’ve been practicing Algebra 1 word problems.
- CueThink provides a space for students to video-record their strategies and interact with others’ work. They can annotate or comment on others’ videos, providing constructive feedback on the process.
- These questions are far more open-ended.
While I have not had a chance to work through a CueThink prompt (it’s not yet at our school), I believe that the UPSC structures we’ve developed provide a perfect space to prepare students for the online program. After practicing so many UPSC examples, my ESOL students should be able to access the math portion of the program, thus enabling them to focus their efforts on understanding the video recording and technology aspects of the assignment.
Good luck! I hope this provides some inspiration for you. 🙂
By the way, you didn’t really think math was *all* gibberish, did you??
– Miss Elsie
Clip Art Credit: https://www.teacherspayteachers.com/Store/The-Hazel-Owl