Math Magic or Calculators?

I was perusing my news feed trying to find something of interest to write about, and came across an article entitled “The Common High School Tool That is Banned in College” i.e. the calculator. It’s an interesting article, worth a read,  basically comparing the high school perspective on the use of calculators to the college perspective or non-use of calculators. There is no right or wrong answer – I think it depends on the math content, what you want students to do (i.e. basic algorithms to solve problems or using mathematics to solve deeper problems).  Depending on your goals, the use of calculators and technology differs. As with any technology, calculators are a resource that needs to be used appropriately, and we need to be teaching that.  Common Core Mathematical Practice #5 – Using Appropriate Tools Strategically is all about this. Calculators have their place and are important to help explore and expand mathematical understanding, but we have to help students understand when their use is necessary and not a ‘crutch’, as stated in the article.

This was on my mind obviously, when I then ran across a tweet post by Go!Math Videos @gomathvideos that shared a TedX talk by Arthur Benjamin entitled “Faster than a Calculator”, which naturally sparked my interest and seemed related to the question of should we be using calculators. In the video, Arthur Benjamin has members of the audience use calculators while he does calculations in his head. He then goes on to wow everyone with his math ‘tricks’ (what he calls mathemagics). He ends by doing a 5-digit square calculation by thinking out loud as he ‘solves’ a problem. It’s fascinating – he changes numbers to words to help him solve – he is definitely using his own ‘algorithm’. The video does not answer the question should we be using calculators – but it definitely shows that calculators are just one way to get a solution and it may not always be the fastest. Anyway – just some fun for this last post of 2016. Enjoy!

Wishing everyone a Happy and Safe New Years!

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Math Practice #5: Use Appropriate Tools Strategically

imagesThere has always been a debate over technology use in the math classroom, starting with using calculators vs. not using calculators. That debate is still raging of course, as evidenced in this recent article at Education World entitled Educators Battle Over Calculator Use: Both Sides Claim Casualties”With all the different technologies in addition to hand-held calculators, such as smart phones, tablets, and laptops, the question of when to use technology becomes even more complicated. This is why I love the Common Core Standards of Math Practice #5, Use appropriate tools strategically. This standard says:

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a stock-photo-18488281-yellow-pencilruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

stock-photo-18684426-3d-yellow-measuring-tapeTools includes any type of resource that a student can utilize, be it paper, pencil, computer or calculator, to help them explore and solve problems. In a mathematics classroom there should NEVER be only one option – there should be multiple options of tools that students may choose from, on their own,  to determine which is the most appropriate for the task at hand. This then makes the debate about calculator vs. no calculator or paper-pencil vs. calculator vs. computer a mute point (at least if all those tools are options in a classroom). The key here, as stated in the standard , is that students are familiar with all tools that would be appropriate for a problem-solving situation, and are given the opportunity to make the choices themselves and are able to JUSTIFY why that tool was the most strategic choice.

An example, from my own teaching experience in high school algebra:

I was a ‘traveling’ teacher for a long time, meaning I did not have my own classroom, but traveled, like the students, to each of my next classes. This forced me to be incredibly organized and to have all my materials with me on a rolling cart. I had a ‘basket of tools” as I called it -fully of calculators, pencils, rulers, graph paper, measuring tapes, compass, protractors, paper clips etc.  Whatever might be needed in the classes I taught (Algebra & Geometry).This basket was always available, and students could get up and retrieve what stock-illustration-7630431-protractor-ruler-measure-angle-lengththey needed, or thought they needed, to support their learning. I did a lot of collaborative learning, so let’s say we were collecting data in class – i.e. measuring body-part lengths for example – and then comparing ratios (Golden Ratio perhaps?) to see if there were any patterns. Some groups started with recording everything on paper-pencil. Some went right to the calculator and the table. Others measured with rulers, realized a measuring tape was more appropriate and switched. When doing the ratios, most reverted to the calculator vs. in their head or paper/pencil because they could quickly get the ratios and begin comparing and looking for patterns. There were lots of options, and each group chose their own approach and used the tools that to them helped them get to the real thinking and application of the problem. If I had only provided what I might have considered the most appropriate tools, measuring tape & calculators, I would have missed the students who started with the rulers and said – wait, this is not going to work because we need something that can bend since our arms are not straight. Or those who said – wait, we can use the table in the calculator to do a quick calculation of the ratio and then see if we can see the pattern. Or those who said lets record it all on paper so we can compare all of us together easier. These were seemingly innocent, obvious comments but demonstrate students thinking about the task, analyzing, collaborating and choosing the most appropriate tools and strategies for them. And THAT is what helps students make sense of learning and understand mathematics.

The question shouldn’t be calculator or no calculator, it should be what tool is going to be the best for to find a solution? Sometimes it might be a calculator. Sometimes it might be mental math. Sometimes it might be a quick drawing with paper & pencil. Students need to always be given the opportunity to choose the most appropriate tool that helps THEM persevere and solve a problem.