Mathematics education

Hmework – A Key to Improvement in Mathematics Education

May 17th, 2017


The United States has a serious problem in mathematics education. Test results show that beginning in middle school, the United States student proficiency declines in comparison with other developed countries throughout the world to near the bottom by 12th grade [PISA]. The Workforce/Education Subcommittee of the President’s Council of Advisors on Science and Technology identified two principal reasons for this: too many math teachers are not trained in math, and too many math textbooks are inadequate [Herbold].

This article suggests a third important contributing reason: low homework emphasis. The United States lags far behind in time spent by our students on homework [TIMMS]. Thus, improving homework completion at the secondary level may be a significant opportunity for US math proficiency improvement at that level. This article suggests that new Internet math homework-help resources may be very helpful in getting more students to complete and understand their homework. As an important by-product, more class time will be available for more in-depth teaching than is currently possible. We identify some of the available resources, discuss their application and the background research that supports their use, present comments from several contributing authors, and present a case study of the use of one of the Internet resources.

Homework Not Being Done

Without homework practice, middle school or high school students cannot fully master the concepts presented in class. Homework provides an opportunity for the students to extend their understanding and directs the students to concepts that need further explanation. Homework provides an opportunity for students to make, discover, and correct mistakes so they can learn from them.

Students of teachers who emphasize the importance of homework score higher than students of teachers who do not [House]. However, teachers are finding that getting students to actually do assigned homework is increasingly difficult. Some students, and often their parents, express that homework is a useless burden. Furthermore, homework competes with TV, video games and other activities. A teacher in Northern California had this to say [Bradley]:

I am a good teacher as evidenced by my experience, by peer, student, and parent feedback, and by advanced certifications that I have earned. Yet, many of my students fail. Most of those who failed will say it was because they did not do the work.

Worked Examples

The use of worked-out examples is a standard practice in classroom lectures and textbooks. Some students understand the daily lesson after observing just a few worked-out examples. Others need more examples than can be presented during a class period or provided in a textbook section. These latter students would benefit significantly from additional worked examples, but they might not know where to find them or have the motivation to seek them out. The research described below shows that providing worked examples for actual assigned homework problems is beneficial. The websites described below allow students to view worked out solutions for their assigned homework.

Internet Resources for Math Homework Help Via Worked Examples

A website that provided math homework-help by showing tutorial (worked out) solutions for actual math textbook homework problems was first introduced in 2001 [Beall]. Other websites with the same general capabilities have since become available. Students needing assistance on assigned homework problems now can find immediate homework help using an Internet-connected computer, at websites such as,,,, and Each of these websites provides immediate tutorial explanations for math problems. Thus, teachers may assign homework problems for which there are explanations (worked solutions) available on the Internet. In the case of Encarta and Hotmath, the explanations are correlated to the actual problems assigned from popular math textbooks.

Even students who may have fallen behind in math may be attracted to such websites, as they provide a new avenue for catching up. The help, via computer, with their assigned homework relieves them of the fear of peer or supervisory embarrassment. Math Teacher Jane Monson noted that more of her students are completing homework since they began using because it answers the question, “What is the next step?” so that students can continue on their own. According to Math Teacher Lisa Winer, “I love that this website gives worked solutions to assigned homework, because students who want it can get instant help right away, on their own. If a student says they didn’t understand the homework, my response is that they should have logged on to get help.”

Scientific research has investigated the use of “worked out” examples in algebra, and the results show that this increases effective learning [Carroll, 1992]. A study was done in Texas: two groups of students, one of poor performers and the other of good performers, were taught together with only one difference; the poor performers were given their homework assignments with 50% of the problems accompanied by worked solutions. The good performers were given the same homework assignments without worked solutions. Interestingly, the poor performers achieved higher scores on the final exam than the original good performers. In related research, students in the worked examples group completed their work more quickly while perceiving the work as less demanding and displayed better performance on tests [Carroll, 1994]. The researchers suggest that the reduced cognitive load allows the students to process the underlying similarity of problems and integrate the methodology with existing knowledge [Grillmeyer, 2001]. They also note that less “wrong learning” results.

The availability of worked solutions also benefits advanced students. They can tackle the more challenging problems with more success and move forward with less outside help. Advanced, motivated students can work ahead in their textbook knowing that an instant tutor is available.
Homework review in class is an important part of math teaching. Teachers need their students to ask questions about processes, rules, and properties as a part of assessing both the students and their own teaching.

Yet, routine questions of interest to only a few students can be a very inefficient use of class time. According to Math Teacher Marty Atkins, “As our students have begun to use ? I am beginning to get more ‘Why did they take that step?’ questions rather than ‘How do I start?’ questions.” A survey of math teachers using one of the homework-help sites concluded that about 20 minutes per class time was freed up [Grillmeyer, 2004]. According to Math Teacher Paula Evans, “We saved class time, which we immediately reallocated to activities which allowed students to develop insight about the material? We have used this time to develop in-class activities which ask students to extend their homework.”

When teachers assign homework problems with solutions available on any of the listed Internet homework help websites, their students can receive step-by-step explained solutions to their actual homework problems. Students are able to see their mistakes and learn from them, and parents are in a better position to see the methods being taught so they can amplify them as needed. Use of these sites is not simply checking or getting answers, but may be considered a directed, self-paced, tutorial experience.

Teachers justifiably want to balance the amount of available homework help so that students are certain to be challenged. Some students might mindlessly copy down solutions if they are available for every problem. The research concludes that 50% of assigned problems should have the available help. Two of the websites that provide solutions to actual textbook homework problems (Encarta and Hotmath) only explain the odd-numbered problems for which numerical answers are already available in the back of the textbook.

The Importance of Good Exams to the Development of Mathematics Education

May 10th, 2017

Tests are supposed to measure the students’ abilities and potential. The student who performs well on a test is supposed to be good or talented in the subject and the student who does not do well in the test is supposed to be poor in the subject and needs extra training.

What is really happening today is that tests do not reflect the students’ abilities. This has disastrous consequences. When unqualified students go to do a mathematics degree and eventually become mathematics teachers or researchers they lead to the decline of future generations in mathematics.

What is happening in schools today is that administrators are applying pressure on teachers and the teachers pass this pressure on to the students. The teachers teach the students only to prepare them for the test. The point is not to make the students understand the concepts. After all this pressure exerted on the teacher he only cares that the students score high on the test. One of the main issues that face the teachers when they do that is that parts of the syllabus are skipped on the test and thus left out untaught by the teacher. However, though these parts that are skipped do not come in the exam they are important for understanding the other parts of the syllabus. This is enough to cripple the students’ understanding of the material. The teachers teach the students some tricks and mechanical drills that allow them to automatically solve exams without understanding.

Tests will be driving standards and curricula in the near future. This is why special attention should be given in how to design tests. In other words, there should be precise definitions of the concrete objectives of the subject and how to quantify the measure of success.

This problem has two sides. One side is the tests and this side is the dominant side due to pressures on schools to have good scores. The second side is the teachers. To drive the teachers to really get the students to understand rather than do mechanical work the exams should be designed to filter the students’ capabilities.

The test should be composed of several sections that cover all the different parts of the syllabus. Each section should contain questions that reflect the students’ abilities in using the different techniques taught. The student should know which technique suitable for which problem. The section should contain problems that examine the students’ abilities in using mathematics in solving real life problems. After all how good is mathematics if one does not know how to apply it to real life.

Part of the exam should be a project on an application of mathematics in real life. The student should use his earned mathematics skills to solve some real life problem in a project. The project should have two supervisors, one supervisor from the student’s school and one supervisor from another school. The students should be exposed to the many innovative software calculators that they can use to complete their projects.

When exams reflect the true abilities of students mathematics teachers would not concentrate on how to beat the system and would concentrate on developing the students’ abilities in the subject.

The school administration should help in this by giving teachers strong support by letting them attend training classes in mathematics and teaching psychology.

The Role of Every Parent in Children Mathematics Education

May 3rd, 2017

Over the years, it has been acceptable to say or tell someone that you can’t do math. The society welcomes it and in many homes today, the saying still goes on. Imagine for example here in the United Kingdom, English is what we speak and you would surely expect everyone you meet in the street or in the office to know the language. It will come as a surprise to you if you happen to ask for direction in the middle of Oxford street London and only to be told ‘sorry, I can’t speak English.’

The role of every parent in children mathematics education is huge. In my professional duty, I do communicate with parents on a regular basis. Personally it does not surprise me when people say that they can’t do math but what worries me is the legacy the impact of the phrase ‘can’t do math’ will have on their children and there after.

Mathematics is a broad subject. Its beauty overlaps in many other subjects and most practically in our daily business activities. This message really needs to be drilled down to the minds of your children. Now, you can promote mathematics in your home by applying the following strategies.

(1) Using your shopping receipts.
After shopping, throwing away the receipt is not helpful by any means particularly with young children at home. Pass the shopping receipts to them and ask them to re-calculate the totals including the deductions. Take away calculator from them and ask to see the working out. This is a good way to set homework for your own children. By doing the long list of calculation, your children will be mapping their minds to helping their parents against over-payment without realising they are practising accounting and computation.

(2) Using the telephone bill.
Please be cautious here. Ideally your children would not interfere with your telephone numbers but, if that is not the case, then block the numbers by crossing them off or doing photocopies. Again, the mathematics to be tapped is huge. Accuracy is number one. Place values and working with decimals are difficult topics with young students, but you see, your children can consolidate on these by working on your telephone bill. Again take away all calculators. The role of every parent in children mathematics education has to be something to be proud of.

(3) Using Gas bills and Electricity bills.
Do the same here. This requires a little bit of high level maths but let them do it. If they can do it once, they will not ask for your help next time round. There is lots of mathematics to be learned from these bills. Skills in conversion is needed, percentages, ratio, meter reading and value added tax calculations are all part of everyday mathematics. By engaging your children to the family mathematics, you are technically equipping them with life essential skills.

The added bonus is that your children will be aware of ways to reduce cost by using less water, switching off lights when not in use, saving food rather than throwing them all away in the bin and guess what, you will be helping them to become more energy efficient thereby saving the environment.