Women in STEM – Celebrating Women’s History Month

Yesterday it was announced that mathematician Karen Uhlenbeck . from the University of Texas at Austin, had been awarded the Abel Prize 2019 “for her pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics.” Impressive in itself, but more impressive because she is the first woman ever to be awarded the prize (The Abel Prize was established on 1 January 2002. The purpose is to award the Abel Prize for outstanding scientific work in the field of mathematics. The prize amount is 6 million NOK (about 750,000 Euro) and was awarded for the first time on 3 June 2003).  A fitting tribute and accomplishment during this month, which happens to be Women’s History Month, which celebrates women’s’ contributions to society and history.

Seems only appropriate to dedicate this post to other significant women and their contributions to STEM, especially as there is still such a need for more women in the STEM fields of science, technology, engineering and mathematics. The more young girls and women see what others have done, the more they are inspired to pursue futures in these fields. I’ve done a little research and pulled together a few names to share in this post. By no means is this an exhaustive list, rather a list of women that sparked my interest, particularly in mathematics, since this has been my personal passion for most of my life. There are many more out there, but the idea of celebrating Women’s History Month is to realize how important, and often unknown/hidden, women have been in many of our STEM advances and historical events.

1. Marie Curie the only woman to have received TWO Nobel Prizes (one for Physics and one for Chemistry).
2. Gertrude B. Elion another Nobel Prize winner in Physiology, whose work contributed to many new drugs, including AZT, the aides drug
3. Augusta Ada King-Noel, Countess of Lovelace – credited with being the first computer programmer!!  Very cool.
4. Barbara McClintock – Nobel Prize winner in Physiology, credited with showing that genes turn certain physical attributes on and off.
5. Rachel Carson – credited with creating the EPA (Environmental Protection Agency) as a result of her writings and work.
6. Radia Perlman – commonly referred to as ‘the Mother of the Internet” for her algorithm (STP) that basically allows the Ethernet to handle massive networks
7. Rear Admiral Grace Hopper – credited with creating the programming language C.O.B.A.L
8. Lisa Meitner – part of the duo that discovered nuclear fission (fascinating history here about her being ignored in the awarding of the Nobel Prize)
9. Katherine Johnson – her mathematical computations influenced every major NASA space project – wow!! (See the movie Hidden Figures)
10. Florence Nightingale – helped pioneer the field of applied statistics and created a version of a pie chart called the ‘coxcomb‘. Totally new information for me!!

I could go on and on – it is amazing once you start looking, how many women have been pioneers, ‘firsts’, and influencers/contributors to math, science, engineering and technology. It’s exciting that so many are finally being recognized. Inspirational. There are lots of interesting articles and synopses out there that can spark student interest and maybe inspire some of our youth as well. Maybe spend some of this Women’s History Month exploring with your students or just on your own. I know I have been really surprised and amazed and plan to keep researching.

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President’s Day – Sample Lesson (Statistics and Measures of Center)

In honor of President’s Day this coming Monday, it seems appropriate for this months’ lesson feature to be center around information about presidents. There is a lot of data that you could explore with students about presidents – age, number years in office, age when they died, number of children, etc. But, I simply chose one statistic – i.e. age at inauguration, to make this really a focused lesson on using a set of data to explore data representations and measures of center.

The Lesson

The idea of this lesson is to help students look at different ways of representing data (table, dot plot, histogram and box-and-whisker plot) in order to make some conjectures and observations. And depending on the representation, really thinking about the information you are able to glean. Can you estimate measures of center from visual representations of data? Is one representation better than another for a given set of data.

ClassPad.net is a great tool for a lesson like this because it allows for students to see all representations in one spot,  continually add on representations, and also provides a place to write down observations and conclusions. You will see that the shared paper (the complete lesson) has explanations, questions, directions all in the one place, and students are doing the math and making their conjectures after each step. It’s an all-in-one, multiple representation activity. Here is the link to the shared paper, that you can use freely with students. If you want to create a duplicate copy to save, you must create a free Classpad.net account and ‘duplicate’ the paper there once you open it. This will create your own copy that you can modify and then share with your own students. Either way, the idea is that students can do multiple representations and measures of central tendency very quickly and make some mathematical connections and conclusions about the given presidential data.

ClassPad.net Lesson In Action

I’ve made a short video that explains the lesson a bit more and walks through the how-to’s of creating the different plots and shows the different aspects of the lesson (i.e. the text stickies, tables, plots, etc.). As usual, you will notice a lot of questions built into the activity, where students are asked to observe, notice, compare and write down their observations. My suggestion for this activity is to do each step and really pause after to allow students individual think time, but then also bring them back together (pair up first, then whole class) to make sure everyone is on the same page before moving to the next step.

Again – here is the shared paper link: ES/MS President’s Day Lesson

Weather and Integers – The Importance of Real World Connections

A lot of my math teacher friends have been posted images from weather reports on FaceBook and Twitter, like this one to the left from @seemathrun, showing the real-world application of integers due to the extreme weather conditions that are happening across the country right now. It really is a perfect opportunity to show a true application of mathematics that students can definitely relate to, especially if they are in those freezing climates. Add in the wind chill, and you have some interesting data and comparisons and a chance to talk about the relevance of math and understanding numbers.  Here’s an image to the left showing wind chill, temperature, and frost bite times someone else shared that could help explain why so many schools are closed, even though there may not be any snow on the ground, (which is usually the reason behind winter closures). I know one of my colleagues and friends, @ClassPadnut, was sharing with me yesterday that with the wind chill, it was -60 where he lives.  Yikes!!!

There is obviously a lot of different math concepts you could explore with students, dependent on grade level and questions asked. I find the wind chill graph the most interesting. Looking at the wind chill chart, the drop in temperature is almost, but not quite, constant, like you would think – i.e. You will note that there is an equation for the calculation of wind chill at the bottom of the image. I was  curious about whether students could find that connection from the data alone -something to challenge students with. How would they graph this data? Could they? Thinking of statistical tables, what would they enter and what statistical plots would be appropriate? If students are in areas where schools actually closed, you could talk about how the data supports the decisions, and what is the ‘cut-off’ temperature/wind speed that might influence the decision? Lots of things to explore.

I found another image that showed the lowest temperatures reported in each state, so you could do a comparison across states. Even Hawaii is cold!!!  Crazy.  Below is the image, which I then used to enter the data in a table in ClassPad.net, and then make two different plots to represent the data – a histogram and a box-plot. You can see from the box plot five-number summary that the median temperature in the U.S. for this day in January is -40.  Wow!!! (And boy, don’t want to be in Alaska at -80!) Again – think of the interesting class discussions about integers, about how these temperatures will impact things such as the orange crops in Florida or the tourism in Hawaii or California. (Here’s a link to the Classpad.net paper that has the image, table, and graphs shown below: https://classpad.net/classpad/papers/share/b61b70a0-0eed-47da-947a-580e1d835f8d.

As you can see, using what is actually happening right now in our country, i.e. REAL world connections of weather (temperature, wind speed, wind chill), is an amazing opportunity to help students see the relevance of integers and statistics and how this data is being used to make important decisions, such as do we close schools? Who should not venture outside? How long before you get frostbite? The visuals help students ‘see’ mathematics in action, and particularly if we focus on the integer aspect, provide a clear connection to integer addition (and subtraction, depending on the questions asked), something many students struggle with.

Whenever possible, we should be trying to connect the math concepts students are learning and using to a real-world application. Here’s a perfect opportunity, no matter the grade level, to have some great class discussions about the impact of weather on our world, about the relevance of integers, and about how statistical information is important to decision making.

 

Math Hardware versus software – Similarities & Differences with Casio

Students using technology as part of learning math is important because of the extension of learning that is possible, the visual connections, and explorations that become possible as a result of technology. The most common technology students use these days are their phones, tablets, computers, and of course, hand-held devices such as calculators. It all depends where you live, what schools you attend, what’s allowed or not allowed, and also what resources are actually available and understood by both teachers and students. From my own research, some schools/teachers have a multitude of resources, but most schools have limited options. And – even if there are many technology tools available, teachers tend to utilize the tool (s) they are most comfortable with, and that the majority of students have access to. Basically, it comes down to choosing a technology that is going to support the learning and that students and teachers can use relatively efficiently, so that time is not lost to ‘tool logistics’. Often times, again, based on my own research (dissertation), teachers choose tools that may NOT be the best choice for learning because they know how to use it over a much better, more appropriate tool, that they are unfamiliar with or uncomfortable with, so many times better technology tools go unused because of the ‘learning curve’.

What I wanted to use this post for today was to show how Casio has really recognized the ‘learning curve’ issue and tried to keep functionality consistent across handheld models and even in their software, providing intuitive steps and menu options right within the graphing menu itself that alleviate some of that ‘learning new tool functionality’ concerns that teachers and students often face when using technology. Our graphing calculators basically use the same steps, buttons, layout, even from the very basic ones (fx9750) (fx9860), to the more advanced ones (CG50), so if you know one, you know them all. And, even the new software, ClassPad.net, is built along the same lines, though obviously with more features and capabilities.  But there is no ‘searching for menus’ – relatively intuitive no matter the tool. Obviously, as you get into the newer models and then into the software, the functionality and options increase – we go from black-and-white displays to color, we go from intersection points on the graphing calculators to union/intersections on the software. But knowing how to use one tool makes transitioning easy, and if you had students with several different models of the handhelds, you could still be talking about the same steps and keystrokes.

The best way to compare and demo is to show you how to do the same thing on the different models. I’ve chosen to show graphing two inequalities, so that you can see, even on the older models, that shading and intersections occur. But also to show that as you progress into the newer and more powerful tools (i.e. memory capacity, color, larger screens, resolution, etc), allowing for more options and learning extensions.

Here are the two inequalities that are being graphed in each of these short GIF’s:

Each GIF below graphs the two inequalities and finds intersection points of the two graphs. The software extends that to allow for finding the Union and the Intersection of all points.

 

fx9750

fx9860

CG50

ClassPad.net FREE Math Software

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Be sure to check out the free software that does calculating, graphing, statistics and geometry: ClassPad.net.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

New Year’s Resolutions – A Chance to Explore Some Statistics

As I was at the gym this morning, noticing the increase in people that were there, I got to thinking about New Year’s Resolutions. I personally dread the month of January at the gym because inevitably, it is a lot more crowded with all the ‘new memberships’ given as gifts over the holidays, and full of new people who have decided losing weight and getting in shape are on their to-do list for this new year. As someone who hits the gym regularly, this month at the beginning of the year is a bit frustrating because machines are taken, the parking lot is crowded, and my regular routine is often interrupted due to the increase in the number of people. I admire everyone’s new-found commitment and applaud the goal of getting in shape and being healthier – however, my anecdotal evidence over the past several years is that this commitment is short-lived for many.  By February, things tend to get back to normal because, sadly, many of our ‘new years resolution’ folks lose the commitment and stop showing up, allowing the rest of us to get back to our routines.

Which brings me back to my thoughts about New Year’s Resolutions (NYR).

From my own very unscientific observations at the gym, those that made NYR to get in shape, lose weight, etc. usually last about a month – and this is based solely on the increase in people during January, and then the slow decrease in people as the month progresses, to the return to the regular crowd by February (with, granted, a few new ‘regulars’ who stick it out). I wondered, as I was cycling, are there any statistics out there that actually show the follow-through on New Year’s Resolutions – i.e. what were the resolutions made at the beginning of the year, and what was the actual end result at the end of the year?

I was able to find statistics on the most popular NYR made last year (2018)  However, I couldn’t find any follow-up statistics to see how many people in the survey actually stuck to their resolutions, which is what I think would be interesting to explore.

 

 

 

 

 

 

 

 

 

 

 

 

 

I then found another source that listed the 10 most popular NYR’s made for this year (2019).  A lot of the same resolutions, though maybe different priority. Some different ones as well, which could be a factor of many things – i.e. the economy, the political climate, the source of the survey, who was surveyed, etc.

I am curious why there is no follow-up from those that conducted the surveys at the end of the year. It would be fascinating to see what the graphs look like at the end of the year compared to the beginning and why or why not some people dropped off their NYR and some stayed true.  I couldn’t find any ‘proof’ for claims such as “80% of all NYR’s fail by February“, though again, going back to my personal observations, I would agree with this claim. There are definitely a lot of articles about how to ‘keep’ your resolutions, and plenty on why people don’t stick to their resolutions, but no statistics that actually support this claim that I could find. But it would be nice to have some data or evidence that supports observations – which leads me to my final thought on a fun ‘real world’ statistical study that teachers might explore with their students for the remainder of this school year.

During this short week, where school has started up again but students tend to still be in vacation-mode, why not start a long-term study to see if we can get some statistical data about NYR’s? Have students in your class make a list of 3 NYR’s – so some goals they really plan/want to accomplish by the end of the school year. Better yet, pick a specific month and/or date (so May 30 for example). Then, compile the class data to create categories and percentages, similar to the charts above. (My guess is students will have some different things on their top 10 list, which would be interesting in itself). Have students keep a record of their progress towards their goals, and maybe on a monthly basis, do a quick survey on students progress/commitment to their NYR’s.  Then at the proposed deadline, do another survey on the success/failure to see who is still working on their goals and who is not. Obviously it is going to be self-reporting, but it would be interesting, as time goes on, to see who is staying committed, who is not, and more importantly, WHY they are not staying committed if that is the case. Do the class results verify that 80% drop off by February? Is there a common theme for those that do not follow-through on their NYR’s?

I wanted to share this as an idea for teachers who might have made their own NYR to be more creative in their math class. The only NYR I ever made each year was to try at least one new thing in my math classes every month – for me a pretty easy resolution to stick to. I would imagine many teachers do something similar. For those of you who have made NYR, good luck and Happy New Year!

 

Systems of Equations – Sample Lessons and Resources

For this months lesson feature, I am going to focus on Systems of Equations. I chose this topic because I just did a workshop with Algebra 1 teachers in NJ, and this is where they were in their pacing guide, so I am making an assumption that many algebra teachers might also be focusing on this content as well this time of year. I am using a problem from Fostering Algebraic Thinking with Casio Technology in order to provide a real-world problem-solving experience (and I have the resource), but I have altered the problem so that I can utilize the all-in-one capabilities of Classpad.net (tables, graphs, equations, geometry, text).

The Problem

In 2010, there were approximately 950,000 doctors in the United States, and approximately 350,000 of them were primary care doctors. It was estimated that more than 45,000 new primary care doctors will be needed by 2020, but the number of medical school students entering family practice decreased by more than 25 percent from 2002 to 2007. With laws reforming health care, many more people will be insured in the United States. 

For many reasons, including a growing and aging population, the demand for doctors will likely increase in future years. The number of doctors available is also expected to increase. But, due to the high cost of insurance and the fear of malpractice lawsuits, many have predicted that the increase in the number of practicing doctors will not keep up with the increase in demand for doctors.

The table to the right provides data from a study conducted in the state of Michigan. These data approximate the number of doctors that were or will actually be licensed and practicing in Michigan, called the supply, and the number of doctors that were or will be needed by the people of Michigan, called demand.

The question is, will there be enough doctors to provide all the services? The shortage of doctors is a problem that challenges the entire country, not just Michigan.

The Lesson

A shared paper has been created in ClassPad.net called Systems of Equations Help! Not Enough Doctors, which you can access by clicking on the title. The idea behind this problem is to provide a real-world context where students can use tables, graphs, and equations (along with calculations) to create a system of equations. They can solve these using methods such as substitution, elimination, and graphing. Students will also be practicing how to model with mathematics, applying what they know about relationships and being able to create a system of equations that fits the context of the situation in order to find a reasonable solution.

In the activity, there is obviously some focus first on getting students to really understand the problem and what the numbers represent, and then the idea is to have them look for patterns and relationships as they look for a solution. First in the table, then by looking at a scatter plot of the data, where they again try to determine a solution based on a visual. Continuing to look for trends, they use prior knowledge to recognize linear relationships, create equations that model the data, and then graph those equations to find a more precise solution. Then, as a check, they solve their system of equations algebraically. It’s all about multiple representations and helping students see the connections between all the representations, and depending on whether you want a specific, precise answer or just a generalized answer, you might choose a different representation.

ClassPad.net – Lesson In Action

The video below shows the activity and does a brief walk through of some of the components and what it would be like doing the activity from a student perspective. I am a big believer in the think-pair-share approach, so I would suggest having students do the Notice and Wonder individually first, then pair up, then share so that you can make sure that any misunderstandings about the context, and clarification about the numbers is figured out before students start solving. Then I would suggest small groups for working on the problem itself.

Other System of Equation Activities and/or video links

• Algebra I Resource Guide (Systems on page 7)
• NCTM’s Problems of The Week (requires member log-in)  Type in ‘systems’ in the search and there are a ton of excellent problems.
• EngageNY Resources on Systems

 

Slow at Math ≠ Bad at Math

*Note: This is a recycled post from my personal blog.

“Speed ISN’T important in math. What is important is to deeply understand mathematical ideas and connections. Whether you are fast or slow isn’t really relevant.” – Laurent Schwartz, mathematician

If you haven’t seen the video by Jo Boaler and some of her Stanford students entitled “How to Learn Math: Four Key Messages”, you definitely need to. Besides the four powerful messages (which I will list below), it has some great stories and quotes, one of which is the one I have above.  Jo Boaler has done powerful research and written some terrific books on mathematics and learning math (one of my favorites being “What’s Math Got to Do with It?” and the video about these four key messages in math is so interesting.

Here are the four key messages about learning math (I highly recommend you watch the video to clarify and define each message a bit more):

1. Everyone can learn math at high levels
2. Believe in yourself (your beliefs about your abilities actually changes the way your brain learns)
3. Struggle and mistakes are really important in learning math
4. Speed is NOT important

All of these speak directly to the way we still, sadly, often teach and learn mathematics. One that really struck out for me was #4, speed is not important. I remember my own daughters struggling with the timed math tests – i.e. you have a minute to try and solve 100 times tables, or complete as many addition problems as possible. Very stressful, very ridiculous, and to top it off, they were penalized with poor grades if they couldn’t reach the arbitrary goal of “x amount of problems in 1 minute”. It still goes on and students memorize and stress over these timed math drills. Why? It’s ridiculous. If we continue to do this to students, then they begin to believe they are bad at math (see #2 above), which leads to them thinking they can’t learn math (see #1), and therefore leads to them giving up when problems get tough (see #3). A self-fulfilling prophecy.

So – I ask those math teachers out there who continue to put pressure on students to perform mathematical skills in a timed matter, where speed is important – stop. Just stop. Focus on what mathematics should be – understanding why those calculations matter, what they are related to, how they help us solve real-world problems. Help students make connections.

I know I keep coming back to it – but the Common Core Mathematical Practices seem to embody these four key messages. No where in there does it say students have to be able to do ___calculations in _____ minutes. Math is NOT about speed – it’s about the struggle, perseverance, conjectures, connections, and applications that help students solve relevant, real-world problems and see the beauty and need for mathematics.

Check out the video here