I don’t think exposing youngsters to the history and philosophy of mathematics will entice more students to the field. Those who are already uninterested in mathematics or find it too difficult won’t be much interested in its history and philosophy either, whereas some of those who are interested may even lose their interest once they learn how messy and unpredictable mathematical progress actually is. However, the history and philosophy of mathematics may be beneficial to graduate students to give them a proper sense of the fluid and impure nature of their discipline.

I don’t think exposing youngsters to the history and philosophy of mathematics will entice more students to the field. Those who are already uninterested in mathematics or find it too difficult won’t be much interested in its history and philosophy either, whereas some of those who are interested may even lose their interest once they learn how messy and unpredictable mathematical progress actually is. However, the history and philosophy of mathematics may be beneficial to graduate students to give them a proper sense of the fluid and impure nature of their discipline.

I think that it can work. Especialy for students who are oriented in humanities and suppose that maths is nothing more then some technique without any deeper overlap. I just published book on infinity ("The Myth of Infinity") that deals with history and philosophy of maths and physics. The concept of infinity is gate to philosophy of maths and on that concept it is possible to ilustrate the role of deeper philosophical foundation of science.

Mathematics is by no means easy and it is quite difficult to attract youngsters to make a career in the field. In grammar school, at least in Europe, youngsters do participate in school competitions in math but these are oriented towards solving problems and thus training for such competitions force them to think by analogy: the more you solve problems at home to more you have a chance to get one similar to one already solved while competing for a prize. This does not foster creativity. Exposing youngsters to the history and practical applications of math could, instead, develop a passion for the real thing...

Maths are typically presented as static concepts. For those students who gravitate to the humanities I think that humanizing mathematics by teaching about its history imperfection could be beneficial and may even produce more creative thinking about the subject. As one of those types of people I always found the sterile way maths were taught to be a bit alienating.

If you have students, you may consider inviting them to investigate Rene Descartes. Descartes is the father of modern philosophy and was a brilliant mathematician. Descartes invented analytical geometry or Cartesian geometry and introduced skepticism as an essential part of the scientific method. Cartesian coordinates are used virtually everywhere in almost anything relatable. Analytical geometry or Cartesian geometry allowed the conversion of geometry into algebra, and vice versa. That could grab someone's interest.

What is the youth you are referring to? Are we talking about primary school students, high school students or undergraduate students? As Dragos Petrescu has pointed out, mathematics in school is focused on problem solving and, quite obviously, those students with less "analytical skills" get discouraged. I therefore tend to agree with Karl Pfeifer that we will not be able to attract more students to the field by exposing them to the history and philosophy of mathematics. However, if the curricula would be enriched with some interesting historical and philosophical background stories, that might help a bit.

following.

I believe that attracting pupils without special inclinations to science is only done by studying the history of a discipline and revealing its practical impact on the world.

On second thought, the instructor and his/her teaching style might be crucial issues. Teaching history and mathematics can be done in so many different ways...

Thanks a lot for your contributions. I think, base on thoughts, this can be a research topic whereby the responds and views of the youngsters will help us to make a very good conclusion. I, also, think the views of the students of higher institutions of learning will be more appropriate in order to obtain a more acceptable conclusion. My question is, can the views of the students of some universities/ colleges / polytechnics of the same state of a country be enough for the conclusion?

I never was a good student of mathematics, spheric trigonometrie, second differentation was the highest level (learned befor set-theory on school). But I was in different times interested in the different ways to think about the philosophical fundations. Starting with Plato's allegory of the cave (Liniengleichnis) and not ending with the new idealism of Riemann and Cantor. Cantor have same roots in Bolzano, and perhaps also in the tradition Kant, Leibniz and scholastic thinking about infinity. Perhaps there is after the first arabic transmission of the »arabic numerals« a second far east influence about the 18. century (Leibniz and China) and Goethe (India). The next step was the foundation of the numerical series by Frege (I think also on Hume) and the discussions with Cantor, Meinong, Husserl and the late Frege (Der Gedanke). At least I think about the questions about Russell and Meinong on the one hand and about Brouwer and Hilbert. But I am mainly interested on the different ways to discuss the relations between intellectual and imagination. So I meditate about the ideas from Hausdorff to Kants constitutive categories.