Before you say anything, yes, the title of this blog post also happens to be the title of Ben Orlin’s amazing blog. I don’t care. I want it to be the title of this post, too.

Math should make sense. Or at least, you should be able to make sense of math. And any drawings you create along the way should aid in that sense making. And any drawings you encounter drawn by someone else should similarly aid in your sense making.

But what happens when they don’t? What happens when kids are forced to do math with – literally – bad drawings?

A few days ago a friend of mine sent me the following message:

“I am so lost trying to help my 4th grader. Do you have a secret website where I can find a strip diagram cheat sheet? I have never seen anything like this before.”

Yeah, me neither. Because this drawing is crap.

I mean, seriously, where do I begin? The three boxes with 32 in them actually make sense. Everything underneath? Not so much, especially to a 9 or 10 year old.

• I tend to prefer curly braces to bracket off clearly defined amounts, like, say, the total. This looks like someone just dragged it over partway to the right and then went, “Eh, good enough.”
• Then there’s a random gap which technically should represent a quantity of its own since this whole thing is built as a linear model.
• And finally we have that little scratch at the end with a 4 under it. Why is that not a curly brace? Are children supposed to know the difference between quantities represented by curly braces and those by line segments that have a slight curve at the end?

Here, let me fix this.

Can I guarantee that the meaning of this particular strip diagram will jump off the page and make sense to anyone who views it? Of course not. But at least now we have some consistency to the stuff on the bottom and the random gap is removed. At least now a child might be able to notice, “Oh the two numbers on the bottom (m and 4) should add up to the total of the numbers in the three boxes above.”

By the way, I should probably stop here and say: Strip diagrams are a TOOL, not a math skill unto themselves! They are meant to be used as a way to represent relationships so that you have an easier time determining which operation(s) to use. So rather than giving a strip diagram and asking students to write an equation and solve it, why not ask, “You set up the following strip diagram to solve a problem. How could you use the diagram to help you find the unknown value? Describe the steps you would take.” Honestly, I care less about students’ computation accuracy with this particular question than I do their ability to tell me that they would do something like multiply 32 times 3 and then subtract 4.

Unfortunately, this wasn’t the only example my friend sent.

Do we hate children? Do we want to make them cry? Because they have every right to as they try to make sense of these horrendous models.

My loathing is not because I can’t figure these out. I have figured them out. And I hate them. They’re just so cumbersome and confusing. Any mathematical meaning they’re trying to convey is muddled by inconsistencies and disproportionate boxes.

Let me make some attempt to fix this. No promises.

I just couldn’t with choice D. That was just a bad model all around. Sure it’s a wrong answer, but there’s no reason it has be a bad model on top of being the wrong answer.

No wonder parents take to Facebook to vent about math these days. If you’re required to use these materials, and I hope you aren’t, then please, please, please keep them at school. For the love of god, don’t send them home.

By the way, all of the drawings I made for this post can be made fairly easily in Google Drawing or the newest version of Powerpoint. They both include automatic features that help you line up and center your boxes and curly braces. Play around and practice. It is well worth your time, not to mention it’s pretty empowering to be able to create the exact strip diagram, number line, or other image you want to use in math class.

Our students deserve to make sense of math with drawings that make sense. Please do everything you can to ensure the only bad drawings are ones students are making themselves because they’re still figuring all this out. With practice and your help, over time they’ll get better.

## 10 thoughts on “Math with Bad Drawings”

1. howardat58

This is garbage. The least one can do is “Hence or otherwise….”, but that is so old fashioned, and unsuitable for grades 4 and 5.
Generally the whole idea should be “Get your answer however you like”.
But i.m an old fuddy-duddy !

2. math on the edge

Thank you for writing all of this. I really appreciate your thoughts. I often struggle with how to introduce students and teachers to some of this tool. Recently, I was coaching a 6th grade teacher and we were teaching a unit on ratio and proportion. We were trying to move intentionally from the concrete to the abstract. We highlighted how you could see equivalent ratios in pictures –>tables with all values —> tables with only some values (because once you see the multiplicative relationship, you don’t need to know “the ratio that comes right before”.) I struggle with when/how to introduce strip diagrams/bar models/ tape diagrams as a tool. I was thinking Cuisinaire rods would be a great way to start concrete. During the unit, I started thinking a lot about our 4th graders and why 4.OA.1 is so important because it really introduces multiplication as scaling. I am going to make a concerted effort to use Cuisinaire rods with 4th grade teachers in a future learning lab. My questions for you are: is it possible to connect the strip diagram to the the continuum I described above? I like to try to show the connections between all models/tools that we use because I want to see the relationships – that is usually where the big ideas hide. 😉 How would you connect the strip diagram to the table? Also, how/when would you introduce the strip diagram? Thanks again! I learn so much from your blog.

1. bstockus Post author

Thank you! I was thinking about your question about whether it’s possible to connect the strip diagram to the continuum you described. It is! Here’s an example. I created a 2:3 ratio using a strip diagram, basically two boxes in a row on top and three boxes of the same size in a row on the bottom.
* If each box has a value of 1, then the two boxes up top have a value of 2 (1 and 1) and the three boxes on the bottom have a value of 3 (1 and 1 and 1).
* If each box has a value of 2, then the two boxes up top have a value of 4 (2 and 2) and the three boxes on the bottom have a value of 6 (2 and 2 and 2).
* If each box has a value of 3, then the two boxes up top have a value of 6 (3 and 3) and the three boxes on the bottom have a value of 9 (3 and 3 and 3).
And so on.
As you try out different values, you can organize the results into a table. Which just so happens to be a ratio table!
I hope that makes sense. I don’t think I can include pictures in my comment so I can’t make a picture to show you. If it doesn’t make sense, message me on Twitter and I’ll share something there.

3. mathmastersblog

Brian, I love your passion for mathematics education, and I truly enjoy your humor, too! Thank goodness we have you in our district to support us and our teachers, but I feel like this post needs one more thing…an Amen!!

1. bstockus Post author

Thank you, Mandy! It was bugging me so much talking through these problems with my friend. I just had to vent somewhere. So glad I have my blog! 🙂

4. Joe Schwartz

I relate to your anger and frustration with poorly designed and constructed worksheets and other curriculum materials. Teachers have enough on their plates without having to worry that every time they turn the page of their manuals there’s the potential to find something that, as Howard says above, is garbage.

5. Jayne

Where are the labels? How is the student supposed to know what anything means? And don’t we usually have the student create the drawing to aid in their understanding not the other way around?

1. bstockus Post author

Hi Jayne,

You make a great point. Labels can be very helpful for students to make connections between a strip diagram representation and a linguistic representation like a word problem. Students need to practice making those connections. Since the purpose of the strip diagram is to help make sense of relationships, I would like for students to eventually have enough grasp to where they wouldn’t necessarily have to include labels next to every number. They would “get” what the numbers represent as they made their own or interpreted someone else’s strip diagram.

Thank you again for the great point!

6. Mark Schruender

I agree with everything you’re saying about the suffering the students are enduring with having to answer questions that don’t make sense. That being said, your ability to make them make sense is a deep level of thinking that maybe we could expose some kids to through differentiation?