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For me, this was an interesting read for me. I have tried, without much success, to get students in the habit of drawing visualizations to represent math problems before trying to solve them. I think the problem for me was that I could not figure out how to explicitly teach the skill of what visual aids might be useful for to determine the meaning of a given word problem. I think it a level of skill to be able to visualize what a problem is asking you. I think students need to first be able to fully understand the text before the can transfer it into a visual aid. Which is a problem in itself that I still continue to struggle with. However, I would like to try to explicit methods outlined in the article. This approach may help my students be able to better understand what the question before even having to consider the math.

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For me, one of the most striking findings from the reading was that students who had LDs were significantly less likely to utilize visual models in their solving of word problems when compared to gifted students. In fact, in the baseline study the researchers found that one of the students with LD had only "generated one diagram out of a possible 24 opportunities to do so" (550).   With my understanding of how students read, this is not surprising at all.  Students who struggle with reading are often not engaging in the process of visualizing what is happening in the text (whether a narrative fiction story, poem, or a non fiction article) when compared to those who have stronger understandings of the main idea of a text. Thus, it follows that these students would struggle equally with visualizing a math problem.

In our curriculum at SA, we lean heavily towards teaching scholars how to use visual models to understand math problems.When I taught 5th grade math, we spent the first few days of teaching decimals using tenths and hundredths charts, while scholars shaded in the appropriate amount of bars/squares for the decimal they needed to represent. Additionally, before we taught them the process for multiplying or dividing fractions, they needed to learn how to represent them in a model first. Yet despite this emphasis, the majority of our  "emergency" scholars, one of their goals is to use a visual model to interpret the problem before they begin to solve.  Though at first I had trouble understanding the concept of a pictorial model vs. a schematic model, after viewing the examples from the article, I realized that schematic models demonstrate a clear mathematical understanding (such as a number line or a percent bar). Then realized that while we are prioritizing the use of visual models, unless scholars combine those visual models with other strategies and understand WHY they can use them to solve the problem, then the use of the model becomes procedural. Students start to draw models simply because they are told to, rather than understanding how this model will help them.   Again, similarly to reading, when we tell scholars to "jot" and "annotate" in order. to better understand the MI of the text, some begin to write nonsense simply because they know they HAVE to write. These procedures that can help students develop stronger understandings cannot become divorced from conceptual understandings of a topic.

Though I do not teach math currently, I do support the math team during the testing season. Additionally, many of the scholars in my advisory and homeroom are our lowest performing math students. I want to encourage them to use visual models in their math work FIRST before blindly calculating information.

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Eamon Deeley's Response    I really liked this reading. In the teaching math class I am taking, I often talk about how I like to draw math out, as that makes it concrete while numerals often feel made up. 

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Eamon Deeley's Response    I really liked this reading. In the teaching math class I am taking, I often talk about how I like to draw math out, as that makes it concrete while numerals often feel made up. 

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I find visualization strategies particularly helpful for students with LD. While I teach social studies, not math, I am still working to bring more into my classroom.

One way I am doing this is by having students use quick sketches to identify, people places or things when annotating. This makes it easier to cite evidence as the drawings serve as quick visual cues for locating the necessary information.

This is also useful in creating timelines of events or cause and effect diagrams. It allows students to play with the same social studies concepts, without being intimidated by the written word.

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It was interesting to read about this study where researchers measured the impact of effective visual representation instruction on student's ability to solve word problems. When I taught fifth grade math, we introduced fraction multiplication and division using visual models. There was a strong sense to "teach them the right way" and "over teaching them Keep, Change, Flip!" On a philosophical, I agreed with both this - KCF is a mindless procedure that does not show any conceptual understanding. Yet, our students were very resistant to using visual models for fraction multiplication and division. It was not until I taught 6th grade Math the following yet did I realize that the issue for them wasn't that they didn't enjoy using a diagram - it was that the diagram didn't make sense for them. These same students did latch onto more visual ways to solve ratio and proportion problems (about 60% of our students used a ratio table, 30% used a percent bar, and 10% used a non-visual way to solve ratios/proportions). This was similar to the findings in this study, where students who were taught about a particular visual model could not connect it back to solve relevant problems (pg. 550). It was only when they were also taught how to use a diagram to problem solve did those same students employ their diagrams. The situation from 5th to 6th was similar for me. In 5th grade, I often felt obligated to expose my students to visual models, so I always introduced them. But we did not authentically use them in problem solving. On the other hand, in 6th grade, we used ratio tables and percent bars consistently in lessons, helping my students (including a class of 12:1:1 students with learning disabilities and speech/language impairments) use those same strategies independently. Thus, my biggest takeaway from this study is that diagrams can be helpful, but only if we as teachers teach the visual representations as a component of the problem solving process, not merely a supplement to the process.

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As I don't teach math this year, I can only speak to my experiences from last year and how these strategies apply to the content area I teach. Teaching the use of diagrams or how to set them up is a useful tool for students in solving math problems. This strategy can also be used in Science. I use it so create a study guide web.

The use of a visual tool helps organize thoughts and main ideas. I believe it is an important skill to have and give my students. It helps them organize and work through the information and their thoughts.

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Alex Bauer's Response:               An important strategy the article discusses is the importance of visual representation in solving mathematical problems. According to the article, teachers should instruct their students on the benefits of visuals and show them how to use visuals to solve mathematical problems. If the use of visual representations are implemented by students, it "adds greater meaning to a task and leads to a greater likelihood of diagrams' being used in other problem-solving situations" (Davis & Maher, 1997). I agree with the strategy of teaching students to use visual representations to solve mathematical problems. There are many types of visuals that students can utilize in order to help them solve mathematical problems. Some examples are diagrams, symbol arrays, pictures, drawings, number lines, tree diagrams, contingency tables and Venn diagrams. The use of visuals can help students in many ways; they can expose important parts of the mathematical problem that might otherwise be overlooked, they can help with multi-step problems and lastly, they can help students gain a better understanding of the problem making it easier to solve. When I give instructional lessons to my class, I always emphasize the importance of visual diagrams and how they can be very helpful in solving the math problem.  

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Alex Bauer's Response:               An important strategy the article discusses is the importance of visual representation in solving mathematical problems. According to the article, teachers should instruct their students on the benefits of visuals and show them how to use visuals to solve mathematical problems. If the use of visual representations are implemented by students, it "adds greater meaning to a task and leads to a greater likelihood of diagrams' being used in other problem-solving situations" (Davis & Maher, 1997). I agree with the strategy of teaching students to use visual representations to solve mathematical problems. There are many types of visuals that students can utilize in order to help them solve mathematical problems. Some examples are diagrams, symbol arrays, pictures, drawings, number lines, tree diagrams, contingency tables and Venn diagrams. The use of visuals can help students in many ways; they can expose important parts of the mathematical problem that might otherwise be overlooked, they can help with multi-step problems and lastly, they can help students gain a better understanding of the problem making it easier to solve. When I give instructional lessons to my class, I always emphasize the importance of visual diagrams and how they can be very helpful in solving the math problem.  

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The use of visuals can be very beneficial, not just for students with disabilities but for any type of learner as well.

Last year, as a High School Global History teacher, my co-teacher and I used a lot of visuals when it came to introducing new vocabulary terms to our students. This year in Math, I have incorporated that strategy and whenever my co-teacher and I introduce new Math terms to our class, we always accompany the word with a visual in order for our students to understand it better.

In Math class, my co-teacher and I use visual representations such as graphs, diagrams, images, etc. In the article, it provides the readers a diagram (pictorial and schematic) that students can use when solving a word problem. I actually found that diagram to be quite interesting and would like to incorporate it into my Math classroom. A few of my students, not just the one's with IEPs, struggle when it comes to solving word problems for they are not sure as to where to begin. Therefore, having them draw it out may be a good alternative for them when it comes to breaking down the word problem into easier segments.

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Although using visuals includes many different strategies in problem solving, I appreciated that this article talked about the difference between certain ones.  For example, while something like an array can be helpful for students who need a visual reminder when multiplying more complex numbers or polynomials, it functions differently than a visual created to diagram distances between houses in a story or plotting information on a number line to better see the relationships between the information.
I have been trying to incorporate more and more of the latter strategies into my classroom by highlighting positive examples of those created by students under the document camera. By blowing up those strategies, students begin to see the merit behind taking the information and synthesizing it in a different way. Students can become creative with how they visualize or draw out the information which is always cool to see. While this does involve calling out some drawings that lack purpose, it is still a great opportunity to see how students see the information presented to them.

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Respond to Teaching Students With LD to Use Diagrams to Solve Mathematical Word Problems 

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Respond to Teaching Students With LD to Use Diagrams to Solve Mathematical Word Problems 

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How can you incorporate a strategy that you learned from this article into your classroom?  How could some of these problem solving strategies help a student or students in your class?

Respond to 2 of your classmates' responses!