Raise your hand if you've had students who felt personally victimized by stoichiometry!

I know. We’ve all been there. Ugh. The year is cruising on, and there your students are, nailing experiment after experiment, mastering atomic structure and dominating nomenclature. You’re hopeful that success will carry into stoichiometry even though they, on the other hand, have no clue what’s on the horizon. Then along comes dimensional analysis setting the stage for a bait-and-switch of monumental proportions, until WHACK – stoichiometry smacks them right in their sense of accomplishment.

Never mind that great percent yield lab you were looking forward to, now!

You see, some kids catch on to the steps involved with doing stoich, while others get seriously mixed up and flounder. And once the stoichiometry wheels have come off the wagon, it can feel like there’s no going back, especially after you’re hit by exchanges with your students like these:

**Student:** *“Which number is supposed to go on bottom of the conversion factor?”*

**Teacher:** *“Look at the units you need to cancel.”*

**Student:** *“Do we multiply or divide here?”*

**Teacher:** *“Multiply, you’re ALWAYS multiplying by something!”*

**Student:** *“I thought it was H _{2}O; so, why are we using 5 for water in this problem?” *

**Teacher:**

*“5 is the coefficient in the balanced equation, it’s different than a subscript”*

**Student:**

*“Oxygen is the limiting reactant, because there is less of it”*

**Teacher:**

*“No, it’s not always the one there’s less of”*

Oh, it gets worse, and we all know it.

### The Issue at Hand

Discouraging stoichiometry conversations can easily deflate any hope you had that students understood what all the symbols and numbers in an equation physically mean. And that confusion about how to view quantities, especially ratios, as representing physical substances in a reaction, is why they struggle to proportionally reason with them as such. Now, proportional thinking is tough for teens in general. It requires more abstract thinking than they typically have developed when they take chemistry. However, regardless of whether or not students possess the cognitive capacity for proportional thinking, often the seeds of thinking around proportions and ratios are planted in math courses before chemistry teachers even meet their students. In math courses integrating physical meaning with proportional thinking is just not the focus (yeah Algebra, we’re talking about you!) The focus is teaching the calculation steps, formula, oralgorithms. And something critical gets glossed over in that math learning, which then becomes significant in chemistry, a single word: “per.” If chemistry teachers ignore this crucial aspect of students’ prior knowledge, then some students will be destined to struggle with stoich.

### But There Is Good News

Commit right now to adopting “for every” speak in your chemistry class this year, and we’ll walk through how to introduce it and use it with students.

Teaching students the proportional reasoning skills needed for stoichiometry doesn’t have to be that daunting. By adjusting how your students talk about stoich, you will adjust how they think about it; eventually, they’ll proportionally reason in a more effective manner. Language is key to learning and chemistry is often regarded as its own language; so, armed with the right choice of words at your disposal, you could easily create a culture of proportional thinking in your classroom. What I have seen with my own students over the past decade is that, by taking advantage of something called neuro-linguistic programming (NLP), you can easily train students to think proportionally, even if they’re still concrete thinkers – all without elaborate hot-dog-and-bun metaphors, fancy picket-fence templates, or Khan Academy videos. Best of all, it only requires trading one Latin preposition you already use – “per” – for its two-word English meaning: “for every.” Commit right now to adopting “for every” speak in your chemistry class this year, and we’ll walk through how to introduce it and use it with students. When we’re done here, you’ll know exactly how to scaffold student thinking to nail the setup of a stoichiometry problem in no time, and possibly where this cognitive approach to proportional thinking could apply elsewhere in your class.

Ready? Let’s get to it.

### These Two Words Will Change How Students Think About Ratios for the Better

Just like you might not make a big deal about Avogadro’s number, in an effort to keep it from becoming a hangup, don’t make a big deal out of the fact that you’re using the words “for every” over “per.” Once I made “for every” my default way of talking about ratios (as opposed to using “per”) students naturally followed my lead in talking and thinking about proportions. Before you launch into “for every” speak with your students, remember this is not a mere ‘shortcut’ or ‘trick’ to help them with chemistry. This is a shift in the way we talk about ratios to use language that embeds the steps to reason proportionally into it. Introducing “for every” speak to your students doesn’t have to wait until stoichiometry, it can be introduced in any context that uses proportions or ratios (e.g., density, molecular mass, graphing). You will, however, want to establish it before stoichiometry with concrete examples. Keep in mind that the context need not be complex.

I. Introduce that “per” actually translates to “for every”

Step 1:Choose a context to activate prior knowledge of the word “per”

Step 2:Elicit some ratio examples from students to list out for them

Step 3:Ask them to define the word “per” (without using the word itself -- so you don’t get “um, like, miles per hour”)

Step 4:Reveal that the word is Latin, meaning “for every”

Step 5:Have them read the example ratios to each other using the words “for every” instead of “per”

Step 6:Ask them a (simple) proportional reasoning question about the ratio in question to prime their thinking.See, nothing fancy going on here.

By introducing it conversationally and through some pedestrian example of using the word “per,” you can actually focus on the physical meaning of the ratio. A great starting context is vehicle speed (or speed limits).

A target student response about speed would be something like:

“for every hour a car travels, it goes 50 miles.”(This convention has a very specific order of independent variable – dependent variable, that way the “for every” speak can naturally be applied to graphs as well.)

II. Help students to apply the “for every” speak about ratios to reason proportionally with ratios.

Step 1:Pose an application question of the ratio.

Step 2:Elicit responses to your question.

Step 3:Have students explain the thinking that led to their response. (and encourage the use of “for every” in the rationale.

Step 4:Write down the response and explanation using mathematical terms to make their thinking visible and start building connections between language, thinking, and symbolic representations.

Step 5:Point out the mathematical steps that they did to arrive at their answer, without even using “an equation” (per se) so they begin to realize they can proportionally reason.

Step 6:Ask a follow-up question that requires inverse mathematical thinking about the ratio ( and then repeat steps 2-5.)Example teacher questions could include things like: “how far would a vehicle travel, at this rate, if it drove for 2 hours? 4 hours? 6.5 hours?”

A target here would be to get students to be able to say something like:

“for every hour it travels 50 miles, so four hours would be 200 miles; because 4 compared to 1 is four times bigger, the miles would have to be 4 times bigger than 50.”Follow-up questions for inverse thinking could include: “how long did a vehicle travel, at this rate, if it drove 150 miles? 75 miles? 1,100 miles?”

A target here would be for them to say something like:

“for every hour it travels 50 miles; so, 150 miles would be 3 hours, since 150 miles is 3 times bigger than 50 miles, the hours would be 3 times bigger than 1.”Depending on how the responses are worded when students share, you might need to push them to elaborate on the thinking behind their calculation, so that they see a difference from the other question. You could ask “how did you know 150 was 3 times bigger than 50?” (or whichever number) until the use of division is revealed as part of the calculation.

Help students to make connections between the two questions, the inverse relationship between the operations that led to the two responses, and the ratio itself. All of this can later on segue into your choice of dimensional analysis formats once students have a conceptual understanding of the relationships between quantities and skill with proportional reasoning.

III. Move on to some ratios without units and follow the same approach to a thought experiment.

Step 1:Write a simple ratio out (e.g., 2/5 or 1/3)

Step 2:Ask students to read them aloud

Step 3:Coach students to read them using “for every” language (e.g., for every 5 parts, you have 2; or, for every 3 parts, there is 1)

Step 4:Point out that even without units, these ratios (e.g., fractions) can be interpreted as a proportion using “for every” speak

Step 5:Ask them to discuss with a partner how the same thinking used in earlier examples with speed could be applied to ratios without units, like these).

Step 6:Elicit responses to invite a discussion that should end in consensus about the similarity between ratios, regardless of their units, being a way of showing “for every something…something else” and lead them to realize the reasoning potential behind talking about ratios like this.Again, all we are doing here is translating the word “per” from Latin to English. And using “for every” to talk about ratios contains the blueprints for how we can think about ratios and

proportionally reason with them.

IV. Move on to “for every” speak using ratios in a chemistry context.Finally, after you've established “for every” speak with students in a simple context, figure out the chemistry content in which you’ll allow students to apply “for every” language. You could choose anything – density, molar mass, or balancing equations – depending on when you introduce “for every” language into your curriculum.

Think of this step like the transition between unit and non-unit ratios – you're scaffolding them to use their own thinking and reasoning in a chemistry context, so that they can work with the ratios you ultimately need them to navigate.

After all, that’s what you ultimately want here, isn’t it?

Let’s say that you chose molar mass. It might shake out like this:

Step 1:Provide students with some molar masses

Step 2:Ask students to read them aloud using “for every” language

Step 3:Pose questions about the molar masses as you did in earlier examples, like speed.

Step 4:Have students share responses and reasoning to the questions

Step 5:Discuss their answers and record the mathematical steps that they verbalized to make the thinking visible

Step 6:Summarize the connections between the ratios, language used to describe them, and the physical meaning of the quantities.Ideally, students would see the same line of reasoning they used earlier can be applied here, despite differences in units, because they’ll start to understand that the ratios represent physical quantities and the language used to talk about them signifies how to work with those quantitites.

A target here would be to get students to say something like:

“for every 1 mol of carbon dioxide, a sample would have 44g of mass; therefore, 3.72 mol of CO_{2}would have a mass that is 3.72 x 44, which would be 163.68g.”You want to make sure students can always come back to talking about things in proportion using “for every” language, regardless of the context, but especially when they get to stoich. Your students should be able to make “for every” statements about the coefficients in the balanced equation as compared to given/experimental amounts of each substance. That way, they are always maintaining connection to the mole ratios in their calculations and thinking during stoichiometry.

Last, but certainly not least, you want to keep in mind that there is no algorithm here, though it may look procedural. That way, you’ll know that this two-word approach to ratios can be applied in many settings and is simple for students to use. Thinking and talking about proportions using “for every” language is about developing student thinking and giving them a reliable means to reason through their chemistry.

### Putting “For Every” Speak to Work in Your Classroom

Ultimately, answers, calculations, and students’ reasoning are all important to understanding in chemistry, especially when it comes to stoichiometry.

If students are not able to make the connection between what’s physically represented by the numbers and symbols, they run the risk of struggling when asked to compare more complex proportions, like those of the ideal ratios (e.g., coefficients) of a balanced chemical equation to actual experimental amounts.

For students who make it through first-year chemistry, stoich might just be the topic that they talk about “surviving.” And rightfully so. It’s tough. But maybe it doesn’t have to be.

By using intentional language to teach students to think more precisely about ratios using the language of “for every” instead of “per,” then we can get them to proportionally reason more effectively and work through quantitative problems with greater confidence.

At this point, you probably have a pretty clear idea about how to introduce and use “for every” statements with your students, which means you start thinking about how they might apply with more context-rich topics such as:

- Enthalpy
- Gas Laws
- Electrochemistry
- Solution Chemistry

As you’ve hopefully seen here, language is key to the formation of our conceptions, and proportion is one concept that’s key to doing chemistry. If we know that helping students better grasp ratios and proportional thinking can help, and if it only requires a simple adjustment in our language to address that issue, then we can make a difference for our students.

At least, that’s what I found in teaching hundreds of chemistry students and in training hundreds of chemistry teachers, as well. Central to “for every” speak is the notion that a change in our language triggers a change in our understanding, and eventually leads to a change in our behavior. Proportional reasoning is no exception to this principle.

With “for every” speak you’re simply translating the Latin word “per” into English. For the wise investment of two measly words, your students just might find that stoichiometry isn’t so daunting after all.

So why wouldn’t you want to give “for every” statements a try with your class?

**Editor's Note: ***This post was submitted for the 2017 ChemEd X Call for Contributions: Creating a Classroom Culture.*

### References

* Abud, G. G. (2011). "For every" speak: A cognitive approach to teaching proportional reasoning (accessed 8/28/17). Paper presented at the biennial meeting of ChemEd, Kalamazoo, MI.

* Gabel, D. (1999). Improving teaching and learning through chemistry education research: A look to the future (accessed 8/28/17). *Journal Of Chemical Education*, 76(4), 548. doi:10.1021/ed076p548

* Draper, S. W. (2003). Vocabulary as a barrier to learning. Retrieved from http://www.psy.gla.ac.uk/~steve/vocab.html#WS (accessed 8/28/17).

* Cassels, J.R.T. & Johnstone, Alex. (1983). The meaning of words and the teaching of chemistry. Education in Chemistry, 20, 10-11.

* Johnstone, A. H. & Selepeng, D. (2001). A language problem revisited (accessed 8/28/17). Chemistry Education: Research and Practice in Europe, 2(1). 19-29.

* Muzio, E. (2017). Coaching: 5 levels of learning. Retrieved from http://www.managetrainlearn.com/page/5-levels-of-learning (accessed 8/28/17).

* Dilts, R. B. (2016). What Is NLP? Retrieved from http://www.nlpu.com/NLPU_WhatIsNLP.html (accessed 8/28/17).

* Garmston, R. & Wellman, B. (2016). The adaptive school: A sourcebook for developing collaborative groups (3rd ed.). Lanham, MD: Rowman & Littlefield.

* Scarino, A. & Liddicoat, A. J. (2009). Teaching and learning languages: A guide (accessed 8/28/17). Carlton, Victoria, AU: Curriculum Corporation.

* Stewart, L. (2016, February 3). Rethinking stoichiometry (accessed 8/28/17) ChemEd X Blog post.

* Carney, M., Smith, E., Hughes, G., Brendefur, J., & Crawford, A. (2016). Influence of proportional number relationships on item accessibility and students’ strategies. Mathematics Education Research Journal, 28(4), 503-522. doi:10.1007/s13394-016-0177-z

* Ramful, A., & Narod, F. B. (2014, March). Proportional reasoning in the learning of chemistry: Levels of complexity. Mathematics Education Research Journal, 26(1), 25–46. doi:10.1007/s13394-013-0110-7.

## NGSS

Mathematical and computational thinking at the 9–12 level builds on K–8 and progresses to using algebraic thinking and analysis, a range of linear and nonlinear functions including trigonometric functions, exponentials and logarithms, and computational tools for statistical analysis to analyze, represent, and model data. Simple computational simulations are created and used based on mathematical models of basic assumptions. Use mathematical representations of phenomena to support claims.

Mathematical and computational thinking at the 9–12 level builds on K–8 and progresses to using algebraic thinking and analysis, a range of linear and nonlinear functions including trigonometric functions, exponentials and logarithms, and computational tools for statistical analysis to analyze, represent, and model data. Simple computational simulations are created and used based on mathematical models of basic assumptions. Use mathematical representations of phenomena to support claims.

## Comments 9

## Excellent advice

im starting molar mass next week. This article comes just in time. Thank you.

## Glad to hear it, Sean! I hope

Glad to hear it, Sean! I hope you find it helpful. It really made the most visible impact in my classroom when students would work through limiting reactant problems. They'd start to predict "for every 2 mol of X, you need 1 mol of Y...so, for the 1.5mol of X we have, we'd need half as much Y, or 0.75mol Y, but...there's only 0.5mol of Y given in the problem; therefore, Y must be the limiting reactant!" It helped us to make the definition of the limiting reactant clearer to kids: "the reactant that there is less than needed for what's available of the other" and it safeguarded against the age-old 'LR is the one there is less of' and reduced the traditional approach I learned of doing the problem based on each reactant to compare yields.

## Thanks..

Totally agree...I'm making a huge poster to put in my room that has "for every" in it...

## Chad, thank you for the great

Chad, thank you for the great idea to create a poster that anchors students' thinking/speaking around "for every" in your classroom! Do share what you come up with, will ya?

Good luck!

## For every

Great article and will use the phrase "for every" this year. I was inquiring if the phrase could be better utilized for the set up 50 miles for every hour? I like the relationship of independent vs dependent variable that for every hour you travel 50 miles but it makes the ratio set up more difficult.

Thank you for getting us to think, Gary

## Hi Diane, thanks for the

Hi Diane, thanks for the feedback on that idea. You raise a good point. I've known it to be used in either order, so make it fit for the conventions of your classroom. Regardless of how you choose to have them say it, the important thing is that they say all of their ratios according to the same convention. That way, as they reason/calculate proportionally, they're comparing corresponding terms of their ratios appropriately. When you get a nice success story, please share it here in the comments!

## Research Interest

I am a Chem teacher who came to some similar strategies. I am interested in investigating how focus on proportional reasoning can make Chemistry more accessible to students. Your article was very helpful and I would like to hear more from you.

## Re: Research Interest

Hi Cathy, thanks for your interest and for reaching out. You might be interested in checking out my website and blog. If you don't find what you are looking for, you can contact me through that same website. ~Gary

## Brilliant!

This is brilliant, thank you for sharing! I actually just gave a quiz over mole conversions and students struggled somewhat with using conversion factors "right side up." I am going to try this out tomorrow!